12-169. The time rate of change of acceleration is

F12-1. Initially, the car travels along a straight road with a speed of 35 in/s. If the brakes are applied and the speed of the car is reduced to 10 m/s in 15 s, determine the constant deceleration of the car.

F12-2. A ball is thrown vertically upward with a speed of 15 m/s. Determine the time of flight when it returns to its original position.

F12-3. A particle travels along a straight line with a velocity of v = (4r - 3^) m/s, where t is in seconds. Determine the position of the particle when t = 4 s. a = 0 when t = 0.

F12-4. A particle travels along a straight line with a speed v = (0.5/3 8/) m/s, where t is in seconds. Determine the acceleration of the particle when t = 2 s.

F12-5. The position of the particle is given by s = (2r 8r + 6) m. where t is in seconds. Determine the time when the velocity of the particle is zero, and the total distance traveled by the particle when t = 3 s.

F12-6. A particle travels along a straight line with an acceleration of a = (10 - 0.2s) m/s2, where s is measured in meters. Determine the velocity of the particle when s = 10 m if v = 5 m/s at s = 0.

F12-7. A particle moves along a straight line such that its acceleration is a = (4r - 2) m/s2, where t is in seconds. When t = 0, the particle is located 2 m to the left of the origin, and when / = 2 s, it is 20 m to the left of the origin. Determine the position of the particle when t = 4 s.

F12-8. A particle travels along a straight line with a velocity of v = (20 - 0.05s2) m/s, where s is in meters. Determine the acceleration of the particle at s = 15 m.

FI2-9. The particle travels along a straight track such that its position is described by the s-t graph. Construct the v-t graph for the same time interval.

A van travels along a straight road with a velocity described by the graph. Construct the s-t and a-t graphs during the same period. Take s=0 when t=0.

F12-11. A bicycle travels along a straight road where its velocity is described by the v-s graph. Construct the a-s graph for the same time interval.

F12-12. The sports car travels along a straight road such that its position is described by the graph. Construct the v-t and a-t graphs for the time interval 0 < t < 10 s. s(m)

The dragster starts from rest and has an acceleration described by the graph. Construct the \(v-t\) graph for the time interval \(0 \leq t \leq t^{\prime}\), where t' is the time for the car to come to rest.

F12-14. The dragster starts from rest and has a velocity described by the graph. Construct the s-t graph during the time interval 0s;< 15s. Also, determine the total distance traveled during this time interval.

F12-15. If the .vandy components of a particle's velocity are vx = (32/) m/s and vy = 8 m/s, determine the equation of the path y = fix). x = 0 and y = 0 when / = 0.

F12-16. A particle is traveling along the straight path. If its position along the .v axis is .v = (8/) m, where / is in seconds, determine its speed when t = 2 s.

F12-17. A particle is constrained to travel along the path. If .v = (4/4) m, where / is in seconds, determine the magnitude of the particles velocity and acceleration when t = 0.5 s.

F12-I8. A particle travels along a straight-line path y = 0.5.v. If the x component of the particles velocity is vx = (2r) m/s, where / is in seconds, determine the magnitude of the particles velocity and acceleration when / = 4 s.

F12-19. A particle is traveling along the parabolic path y = 0.25.V2. If x = (2/2) m, where t is in seconds, determine the magnitude of the particle's velocity and acceleration when t = 2 s.

The box slides down the slope described by the equation \(y=\left(0.05 x^2\right) \mathrm{m}\), where x is in meters. If the box has x components of velocity and acceleration of \(v_x=-3 \mathrm{~m} / \mathrm{s}\) and \(a_x=-1.5 \mathrm{~m} / \mathrm{s}^2\) at \(x=5 \mathrm{~m}\), determine the y components of the velocity and the acceleration of the box at this instant.

The ball is kicked from point A with the initial velocity \(v_{\mathrm{A}}=10 \mathrm{\ m} / \mathrm{s}\). Determine the maximum height h it reaches.

The ball is kicked from point A with the initial velocity \(v_A=10 \mathrm{~m} / \mathrm{s}\). Determine the range R, and the speed when the ball strikes the ground.

f12-23. Determine the speed at which the basketball at A must be thrown at the angle of 30 so that it makes it to the basket at B.

F12-24. Water is sprayed at an angle of 90 from the slope

F12-25. A ball is thrown from A. If it is required to clear the wall at B, determine the minimum magnitude of its initial velocity vv

F12-26. A projectile is fired with an initial velocity of vA = 150 m/s off the roof of the building. Determine the range R where it strikes the ground at B.

F12-27. The boat is traveling along the circular path with a speed of v = (0.0625t2) m/s, where t is in seconds. Determine the magnitude of its acceleration when t = 10 s.

F12-28. The car is traveling along the road with a speed of v= (2 s) m/s, where s is in meters. Determine the magnitude of its acceleration when s = 10 m.

F12-29. If the car decelerates uniformly along the curved road from 25 m/s at T to 15 m/s at C, determine the acceleration of the car at B.

FI 2-30. When x = 10 ft, the crate has a speed of 20 ft/s which is increasing at 6 ft/s2. Determine the direction of the crate's velocity and the magnitude of the crates acceleration at this instant.

F12-31. If the motorcycle has a deceleration of a, = (0.001 s) m/s2 and its speed at position A is 25 m/s, determine the magnitude of its acceleration when it passes point B.

F12-32. The car travels up the hill with a speed of v = (0.2s) m/s, where s is in meters, measured from A. Determine the magnitude of its acceleration when it is at point s = 50 m, where p = 500 m.

F12-33. The car has a speed of 55 ft/s. Determine the angular velocity 0 of the radial line OA at this instant.

F12-34. The platform is rotating about the vertical axis such that at any instant its angular position is 0 = (4/v2) rad, where t is in seconds. A ball rolls outward along the radial groove so that its position is r (O.lr'tm, where t is in seconds. Determine the magnitudes of the velocity and acceleration of the ball when t 1.5 s.

F12-35. Peg P is driven by the fork link OA along the curved path described by r = (26) ft. At the instant 0 = tt/4 rad, the angular velocity and angular acceleration of the link are 0 = 3 rad/s and 0 = 1 rad/s2. Determine the magnitude of the peg's acceleration at this instant.

Peg P is driven by the forked link OA along the path described by \(r=e^{\theta}\), where r is in meters. When \(\theta=\frac{\pi}{4} \mathrm{\ rad}\) the link has an angular velocity and angular acceleration of \(\dot{\theta}=2 \mathrm{\ rad} / \mathrm{s} \text { and } \ddot{\theta}=4 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the radial and transverse components of the pegs acceleration at this instant.

F12-37. The collars arc pin connected at B and are free to move along rod OA and the curved guide OC having the shape of a cardioid, r = [0.2(1 + cos 0)1 in. At 0 = 30, the angular velocity of OA is 0 = 3 rad/s. Determine the magnitude of the velocity of the collars at this point.

F12-38. At the instant 0 = 45, the athlete is running with a constant speed of 2 m/s. Determine the angular velocity at which the camera mast turn in order to follow the motion.

F12-39. Determine the velocity of block D if end A of the rope is pulled down with a speed of vA = 3 m/s.

FI2-40. Determine the velocity of block A if end B of the rope is pulled down with a speed of 6 m/s.

F12-41. Determine the velocity of block /I if end B of the rope is pulled down with a speed of 1.5 m/s.

F12-42. Determine the velocity of block A if end F of the rope is pulled down with a speed of vh = 3 m/s.

F12-43. Determine the velocity of car A if point P on the cable has a speed of 4 m/s when the motor M winds the cable in.

F12-44. Determine the velocity of cylinder B if cylinder A moves downward with a speed of vA = 4 ft/s.

FI 2-45. Car 4 is traveling with a constant speed of BO km/h due north, while car B is traveling with a constant speed of 100 km/h due east. Determine the velocity of car B relative to car A.

FI 2-46. Two planes A and B are traveling with the constant velocities shown. Determine the magnitude and direction of the velocity of plane B relative to plane A.

F12-47. The boats A and B travel with constant speeds of vA 15 m/s and vH = 10 m/s when they leave the pier at O at the same time. Determine the distance between them when t = 4 s.

F12-48. At the instant shown, cars A and B are traveling at the speeds shown. If B is accelerating at 1200 km/h2 while A maintains a constant speed, determine the velocity and acceleration of A with respect to B.

12-1. A baseball is thrown downward from a 50-ft tower with an initial speed of 18 ft/s. Determine the speed at which it hits the ground and the time of travel.

12-2. When a train is traveling along a straight track at 2 m/s, it begins to accelerate at a = (60 u-4) m/s-, where v is in m/s. Determine its velocity v and the position 3 s after the acceleration.

12-3. From approximately what floor of a building must a car be dropped from an at-rest position so that it reaches a speed of 80.7 ft/s (55 mi/h) when it hits the ground? Each floor is 12 ft higher than the one below it. (Note: You may want to remember this when traveling 55 mi/h.)

Traveling with an initial speed of 70 km/h, a car accelerates at \(6000 \mathrm{\ km} / \mathrm{h}^{2}\) along a straight road. How long will it take to reach a speed of 120 km/h? Also, through what distance does the car travel during this time?

12-5. A bus starts from rest with a constant acceleration of 1 m/s2. Determine the time required for it to attain a speed of 25 m/s and the distance traveled.

12-6. A stone A is dropped from rest down a well, and in 1 s another stone B is dropped from rest. Determine the distance between the stones another second later.

12-7. A bicyclist starts from rest and after traveling along a straight path a distance of 20 m reaches a speed of 30 km/h. Determine his acceleration if it is constant. Also, how long does it take to reach the speed of 30 km/h?

A particle moves along a straight line with an acceleration of \(a=5/\left(3s^{1/3}+s^{5/2}\right)\mathrm{\ m}/\mathrm{s}^2\), where s is in meters. Determine the particle's velocity when s = 2 m, if it starts from rest when s = 1 m. Use a numerical method to evaluate the integral.

If it takes 3 s for a ball to strike the ground when it is released from rest, determine the height in meters of the building from which it was released. Also, what is the velocity of the ball when it strikes the ground?

12-10. The position of a particle along a straight line is given by s = (1.5/3 - 13.5/2 + 22.5/) ft, where t is in seconds. Determine the position of the particle when t = 6 s and the total distance it travels during the 6-s time interval. Hint: Plot the path to determine the total distance traveled.

12-11. If a particle has an initial velocity of v0 = 12 ft/s to the right, at s0 = 0, determine its position when / = 10 s, if a = 2 ft/s2 to the left.

12-12. Determine the time required for a car to travel 1 km along a road if the car starts from rest, reaches a maximum speed at some intermediate point, and then stops at the end of the road. The car can accelerate at 1.5 m/s2 and decelerate at 2 m/s2.

12-13. Tests reveal that a normal driver takes about 0.75 s before he or she can react to a situation to avoid a collision. It lakes about 3 s for a driver having 0.1% alcohol in his system to do the same. If such drivers are traveling on a straight road at 30 mph (44 ft/s) and their cars can decelerate at 2 ft/s2, determine the shortest stopping distance d for each from the moment they see the pedestrians. Moral: If you must drink, please don't drive!

12-14. A car is to be hoisted by elevator to the fourth floor of a parking garage, which is 48 ft above the ground. If the elevator can accelerate at 0.6 ft/s2, decelerate at 0.3ft/s2, and reach a maximum speed of 8 ft/s, determine the shortest time to make the lift, starting from rest and ending at rest.

12-15. A train starts from rest at station A and accelerates at 0.5 m/s2 for 60 s. Afterwards it travels with a constant velocity for 15 min. It then decelerates at 1 m/s2 until it is brought to rest at station B. Determine the distance between the stations.

A particle travels along a straight line such that in 2 s it moves from an initial position \(s_{A}=+0.5 \mathrm{\ m}\) to a position \(s_{B}=-1.5 \mathrm{\ m}\). Then in another 4 s it moves from \(s_{B}\) to \(s_C=+2.5\mathrm{\ m}\). Determine the particle's average velocity and average speed during the 6-s time interval.

The acceleration of a particle as it moves along a straight line is given by \(a=(2t-1)\mathrm{\ m}/\mathrm{s}^2\), where t is in seconds. If s = 1 m and v = 2 m/s when t = 0, determine the particle's velocity and position when t = 6 s. Also, determine the total distance the particle travels during this time period.

12-18. A freight train travels at v = 60(1 e~')ft/s, where / is the elapsed time in seconds. Determine the distance traveled in three seconds, and the acceleration at this time.

12-19. A particle travels to the right along a straight line with a velocity v [5/(4 + 5)] m/s, where s is in meters. Determine its position when / = 6sifs = 5m when t = 0.

12-20. The velocity of a particle traveling along a straight line is v = (3t2 - 6/) ft/s, where t is in seconds. If s = 4 ft when t 0, determine the position of the particle when t 4 s. What is the total distance traveled during the time interval / = 0 to t 4 s? Also, what is the acceleration when t 2 s?

12-21. If the effects of atmospheric resistance are accounted for, a freely falling body has an acceleration defined by the equation a = 9.8l[l - v2 (10~4)1 m/s2, where v is in m /s and the positive direction is downward. If the body is released from rest at a very high altitude, determine (a) the velocity when / = 5 s, and (b) the bodys terminal or maximum attainable velocity (as t*<*>).

12-22. The position of a particle on a straight line is given by.s' = (r3 -9r + 15/) ft, where t is in seconds. Determine the position of the particle when / = 6 s and the total distance it travels during the 6-s time interval. Hint: Plot the path to determine the total distance traveled.

12-23. Two particles A and B start from rest at the origin s = 0 and move along a straight line such that aA = (61 3) ft/s2 and aH = (12r - 8) ft/s2, where t is in seconds. Determine the distance between them when t = 4 s and the total distance each has traveled in t = 4 s.

12-24. A particle is moving along a straight line such that its velocity is defined as v = (4.v2) m/s, where s is in meters. If s = 2 m when t = 0. determine the velocity and acceleration as functions of time.

12-25. A sphere is fired downwards into a medium with an initial speed of 27 m/s. If it experiences a deceleration of a = (-6/) m/s2, where/isin seconds.determine the distance traveled before it stops.

12-26. When two cars A and B are next to one another, they arc traveling in the same direction with speeds vA and Vy, respectively. If B maintains its constant speed, while A begins to decelerate at a,determine the distance d between the cars at the instant A stops.

12-27. A particle is moving along a straight line such that when it is at the origin it has a velocity of 4 m/s. If it begins to decelerate at the rate of a = (-1.5u ^2) m/s2, where v is in m/s, determine the distance it travels before it stops.

A particle travels to the right along a straight line with a velocity \(v=[15/(4+s)]\ m/s\), where s is in meters. Determine its deceleration when s = 2 m.

*12-28. A particle travels to the right along a straight line with a velocity v = [5/(4 + s)] m/s, where s is in meters. Determine its deceleration when s = 2 m.

As a train accelerates uniformly it passes successive kilometer marks while traveling at velocities of 2 m/s and then 10 m/s. Determine the train’s velocity when it passes the next kilometer mark and the time it takes to travel the 2-km distance.

The acceleration of a particle along a straight line is defined by \(a=(2 t-9) \mathrm{m} / \mathrm{s}^{2}\), where t is seconds. At t = 0, s = 1 m and v = 10 m/s. When t = 9 s, determine (a) the particle's position, (b) the total distance traveled, and (c) the velocity.

12-31. The acceleration of a particle along a straight line is defined by a = (2/ - 9) m/s2, where t is in seconds. At / = 0, s = 1 m and v = 10 m/s. When / = 9 s, determine (a) the particle's position, (b) the total distance traveled, and (c) the velocity.

At t = 0 bullet A is fired vertically with an initial (muzzle) velocity of 450 m/s. When t = 3 s, bullet B is fired upward with a muzzle velocity of 600 m/s. Determine the lime t, after A is fired, as to when bullet B passes bullet A. At what altitude does this occur?

A boy throws a ball straight up from the top of a 12-m high tower. If the ball falls past him 0.75 s later, determine the velocity at which it was thrown, the velocity of the ball when it strikes the ground, and the time of flight.

12-35. When a particle falls through the air, its initial acceleration a = g diminishes until it is zero, and thereafter it falls at a constant or terminal velocity Vr. If this variation of the acceleration can be expressed as a = (gjv2t)(v2t - tr), determine the time needed for the velocity to become v = Vfj2. Initially the particle falls from rest.

12-36. A particle is moving with a velocity of u when 5 = 0 and t = 0. If it is subjected to a deceleration of a = -kiJ, where k is a constant, determine its velocity and position as functions of time.

12-37. As a body is projected to a high altitude above the earths surface, the variation of the acceleration of gravity with respect to altitude y must be taken into account. Neglecting air resistance, this acceleration is determined from the formula a = -g0\R2/(R + y)2], where g0 is the constant gravitational acceleration at sea level, R is the radius of the earth, and the positive direction is measured upward. If g0 = 9.81 m/s2 and R = 6356 km, determine the minimum initial velocity (escape velocity) at w'hich a projectile should be shot vertically from the earths surface so that it does not fall back to the earth. Hint: This requires that v = 0 as y *

Accounting for the variation of gravitational acceleration a with respect to altitude y (see Prob. 12-37), derive an equation that relates the velocity of a freely falling particle to its altitude. Assume that the particle is released from rest at an altitude \(y_0\) from the earth's surface. With what velocity does the particle strike the earth if it is released from rest at an altitude \(y_0=500 \mathrm{~km}\)? Use the numerical data in Prob. 12-37.

12-39. A freight train starts from rest and travels with a constant acceleration of 0.5 ft/s2. After a time /' it maintains a constant speed so that when / = 160 s it has traveled 2000 ft. Determine the time t' and draw the v-t graph for the motion.

*12-40. A sports car travels along a straight road with an acceleration-deceleration described by the graph. If the car starts from rest, determine the distance s' the car travels until it stops. Construct the v-s graph for 0 ^ s < s.

12-41. A train starts from station A and for the first kilometer, it travels with a uniform acceleration. Then, for the next two kilometers, it travels with a uniform speed. Finally, the train decelerates uniformly for another kilometer before coming to rest at station B. If the time for the whole journey is six minutes, draw the v-t graph and determine the maximum speed of the train.

12-42. A particle starts from 5 = 0 and travels along a straight line with a velocity v = (t2 - 4t + 3) m/s. where t is in seconds. Construct the v-t and a-t graphs for the time interval 0 ^ t ^ 4 s.

12-43. If the position of a particle is defined by 5 = |2 sin (tt/5)/ + 4| m, where / is in seconds, construct the s-t, v-t, and a-t graphs for 0 < / < 10 s.

*12-44. An airplane starts from rest, travels 5000 ft down a runway, and after uniform acceleration, takes off with a speed of 162 mi/h. It then climbs in a straight line with a uniform acceleration of 3 ft/s2 until it reaches a constant speed of 220 mi/h. Draw the s-t, v-l, and a-t graphs that describe the motion.

12-45. The elevator starts from rest at the first floor of the building. It can accelerate at 5 ft/s2 and then decelerate at 2 ft/s2. Determine the shortest time it takes to reach a floor 40 ft above the ground. The elevator starts from rest and then stops. Draw the a-t, v-t, and s-t graphs for the motion.

12-46. The velocity of a car is plotted as shown. Determine the total distance the car moves until it stops (/ = 80 s). Construct the a-t graph.

12-47. The vs graph for a go-cart traveling on a straight road is shown. Determine the acceleration of the go-cart at s = 50 m and s = 150 m. Draw the as graph.

The v-t graph for a particle moving through an electric field from one plate to another has the shape shown in the figure. The acceleration and deceleration that occur are constant and both have a magnitude of \(4 \mathrm{\ m} / \mathrm{s}^{2}\). If the plates are spaced 200 mm apart, determine the maximum velocity \(v_{\max }\) and the time \(t^{\prime}\) for the particle to travel from one plate to the other. Also draw the s-t graph. When \(t=t^{\prime} / 2\) the particle is at s = 100 mm.

12-49. The v-t graph for a particle moving through an electric field from one plate to another has the shape shown in the figure, where i' = 0.2 s and i>max = 10 m/s. Draw the s-t and a-t graphs for the particle. When / = t' j2 the particle is at s = 0.5 m.

12-50. The v-t graph of a car while traveling along a road is shown. Draw the s-t and a-t graphs for the motion.

12-51. The a-t graph of the bullet train is shown. If the train starts from rest, determine the elapsed time t' before it again comes to rest. What is the total distance traveled during this time interval? Construct the v-t and s-t graphs.

12-52. The snowmobile moves along a straight course according to the v-t graph. Construct the s-t and a-t graphs for the same 50-s time interval. When / = 0, s = 0.

12-53. A two-stage missile is fired vertically from rest with the acceleration shown. In 15 s the first stage A burns out and the second stage B ignites. Plot the v-t and s-t graphs which describe the two-stage motion of the missile for

12-54. The dragster starts from rest and has an acceleration described by the graph. Determine the time t' for it to stop. Also, what is its maximum speed? Construct the v-t and s-t graphs for the time interval 0

12-55. A race car starting from rest travels along a straight road and for 10 s has the acceleration shown. Construct the v-t graph that describes the motion and find the distance traveled in 10 s.

*12-56. The v-t graph for the motion of a car as if moves along a straight road is shown. Draw the a-t graph and determine the maximum acceleration during the 30-s time interval. The car starts from rest at s = 0.

12-57. The v-t graph for the motion of a car as it moves along a straight road is shown. Draw the s-t graph and determine the average speed and the distance traveled for the 30-s time interval.The car starts from rest at s 0.

12-58. The jet-powered boat starts from rest at s = 0 and travels along a straight line with the speed described by the graph. Construct the s-t and a-t graph for the time interval 0 ^ ^ 50 s.

12-59. An airplane lands on the straight runway, originally traveling at 110 ft/s when s = 0. If it is subjected to the decelerations shown, determine the time /' needed to stop the plane and construct the s-t graph for the motion.

*12-60. A car travels along a straight road with the speed shown by the v-t graph. Plot the a-t graph.

12-61. A car travels along a straight road with the speed shown by the v-t graph. Determine the total distance the car travels until it stops when / = 48 s. Also plot the s-t graph.

12-62. A motorcyclist travels along a straight road with the velocity described by the graph. Construct the s-t and a-t graphs.

12-63. The speed of a train during the first minute has been recorded as follows: t (s) 0 20 40 60 v (m/s) 0 16 21 24 Plot the v-t graph, approximating the curve as straight-line segments between the given points. Determine the total distance traveled.

*12-64. A man riding upward in a freight elevator accidentally drops a package off the elevator when it is 100 ft from the ground. If the elevator maintains a constant upward speed of 4 ft/s, determine how high the elevator is from the ground the instant the package hits the ground. Draw the v-t curve for the package during the time it is in motion. Assume that the package was released with the same upward speed as the elevator.

12-65. Two cars start from rest side by side and travel along a straight road. Car A accelerates at 4 m/s2 for 10 s and then maintains a constant speed. Car B accelerates at 5 m/s2 until reaching a constant speed of 25 m/s and then maintains this speed. Construct the a-t, v-t, and s-t graphs for each car until t = 15 s. What is the distance between the two cars when t = 15 s?

12-66. A two-stage rocket is fired vertically from rest at s = 0 with an acceleration as shown. After 30 s the first stage A burns out and the second stage B ignites. Plot the v-t graph which describes the motion of the second stage for 0 s t : 60 s.

12-67. A two-stage rocket is fired vertically from rest at s = 0 with an acceleration as shown. After 30 s the first stage A burns out and the second stage B ignites. Plot the s-t graph which describes the motion of the second stage for 0 ^ t 60 s.

12-68. The a-s graph for a jeep traveling along a straight road is given for the first 300 m of its motion. Construct the v-sgraph.At5 = 0, v = 0.

12-69. The v-s graph for the car is given for the first 500 ft of its motion. Construct the <7-5 graph for 0^5^ 500 ft. How long does it take to travel the 500-ft distance? The car starts at s = 0 when 7 = 0.

12-70. The boat travels along a straight line with the speed described by the graph. Construct the s-t and a-s graphs. Also, determine the time required for the boat to travel a distance s = 400 m if s = 0 when / = 0.

12-71. The v-s graph of a cyclist traveling along a straight road is shown. Construct the a-s graph.

12-72. The a-s graph for a boat moving along a straight path is given. If the boat starts at s = 0 when v = 0, determine its speed when it is at s = 75 ft, and 125 ft. respectively. Use a numerical method to evaluate v at s = 125 ft.

The position of a particle is defined by \(r=\{5(\cos 2 t) \mathbf{i}\ +\ 4(\sin 2 t) \mathbf{j}\}\ \mathrm{m}\), where t is in seconds and the arguments for the sine and cosine are given in radians. Determine the magnitudes of the velocity and acceleration of the particle when t = 1 s. Also, prove that the path of the particle is elliptical.

12-74. The velocity of a particle is v = { 3i + (6 2/)j } m/s, where t is in seconds. If r = 0 when / = 0. determine the displacement of the particle during the time interval t = 1 s to t = 3 s.

12-75. A particle, originally at rest and located at point (3 ft, 2 ft, 5 ft), is subjected to an acceleration of a = {6/i + I2rk} ft/s2. Determine the particles position (x, y, z) at t = I s.

*12-76. The velocity of a particle is given by v { 16ri +4rj + (5/ + 2)k } m/s, where / is in seconds. If the particle is at the origin when / = 0. determine the magnitude of the particles acceleration when t = 2 s. Also, what is the x, y, z coordinate position of the particle at this instant?

12-77. The car travels from A to B, and then from B to C, as shown in the figure. Determine the magnitude of the displacement of the car and the distance traveled.

12-78. A car travels east 2 km for 5 minutes, then north 3 km for 8 minutes, and then west 4 km for 10 minutes. Determine the total distance traveled and the magnitude of displacement of the car. Also, what is the magnitude of the average velocity and the average speed?

12-79. A car traveling along the straight portions of the road has the velocities indicated in the figure when it arrives at points A, B, and C. If it takes 3 s to go from A to B, and then 5 s to go from B to C, determine the average acceleration between points A and B and between points A and C.

12-80. A particle travels along the curve from A to B in 2 s. It takes 4 s for it to go from B to C- and then 3 s to go from C to D. Determine its average speed when it goes from A to D.

12-81. The position of a crate sliding down a ramp is given by x = (0.25/3) m.y = (1,5/2) m, z = (6 - 0.75lsn) m, where t is in seconds. Determine the magnitude of the crate's velocity and acceleration when t 2 s.

12-82. A rocket is fired from rest at x = 0 and travels along a parabolic trajectory described byy'2 = [120(103).v] m. If the x component of acceleration is ax = (-/2) m/s2, where t is in seconds, determine the magnitude of the rockets velocity and acceleration when t 10 s.

12-83. The particle travels along the path defined by the parabola y 0.5.v2. If the component of velocity along the x axis is vx = (5/) ft/s, where t is in seconds, determine the particle's distance from the origin O and the magnitude of its acceleration when r = I s. When / = 0, x = 0, y = 0.

12-84. The motorcycle travels with constant speed v() along the path that, for a short distance, takes the form of a sine curve. Determine the x and y components of its velocity at any instant on the curve.

12-85. A particle travels along the curve from A to B in 1 s. If it takes 3 s for it to go from A to C, determine its average velocity when it goes from B to C.

12-86. When a rocket reaches an altitude of 40 m it begins to travel along the parabolic path (y - 40)2 = 160.V, where the coordinates are measured in meters. If the component of velocity in the vertical direction is constant at vr = 180 m/s, determine the magnitudes of the rockets velocity and acceleration when it reaches an altitude of 80 m.

Pegs A and B are restricted to move in the elliptical slots due to the motion of the slotted link. If the link moves with a constant speed of 10 m/s, determine the magnitude of the velocity and acceleration of peg A when x = 1 m.

The van travels over the hill described by \(y=\left(-1.5\left(10^{-3}\right) x^2+15\right) \mathrm{ft}\). If it has a constant speed of \(75 \mathrm{ft} / \mathrm{s}\), determine the x and y components of the van's velocity and acceleration when \(x=50 \mathrm{ft}\).

12-89. It is observed that the time for the ball to strike the ground at B is 2.5 s. Determine the speed vA and angle 0A at which the ball was thrown.

12-90. Determine the minimum initial velocity % and the corresponding angle 0O at which the ball must be kicked in order for it to just cross over the 3-m high fence.

12-91. During a race the dirt bike was observed to leap up off the small hill at A at an angle of 60 with the horizontal. If the point of landing is 20 ft away, determine the approximate speed at which the bike was traveling just before it left the ground. Neglect the size of the bike for the calculation.

12-92. The girl always throws the toys at an angle of 30 from point A as shown. Determine the time between throws so that both toys strike the edges of the pool B and C at the same instant. With what speed must she throw each toy?

12-93. The player kicks a football with an initial speed of Vo = 90 ft/s. Determine the time the ball is in the air and the angle 6 of the kick.

12-94. From a videotape, it was observed that a player kicked a football 126 ft during a measured time of 3.6 seconds. Determine the initial speed of the ball and the angle 0 at which it was kicked.

12-95. A projectile is given a velocity v0 at an angle <f> above the horizontal. Determine the distance d to where it strikes the sloped ground. The acceleration due to gravity is g.

12-96. A projectile is given a velocity v0. Determine the angle <f> at which it should be launched so that d is a maximum. The acceleration due to gravity is g.

12-97. Determine the minimum height on the wall to which the firefighter can project water from the hose, so that the water strikes the wall horizon tally. The speed of the water at the nozzle is % = 48 ft/s.

12-98. Determine the smallest angle 6, measured above the horizontal, that the hose should be directed so that the water stream strikes the bottom of the wall at B. The speed of the water at the nozzle is Vc = 48 ft/s.

12-99. Measurements of a shot recorded on a videotape during a basketball game are shown. The ball passed through the hoop even though it barely cleared the hands of the player B who attempted to block it. Neglecting the size of the ball, determine the magnitude vA of its initial velocity and the height h of the ball when it passes over player B.

12-100. It is observed that the skier leaves the ramp A at an angle 0A = 25 with the horizontal. If he strikes the ground at B, determine his initial speed vA and the time of flight iAB.

It is observed that the skier leaves the ramp A at an angle \(\theta_{A}=25^{\circ}\) with the horizontal. If he strikes the ground at B, determine his initial speed \(v_{A}\) and the speed at which he strikes the ground.

A golf ball is struck with a velocity of 80 ft/s as shown. Determine the distance d to where it will land.

12-103. The ball is thrown from the tower with a velocity of 20 ft/s as shown. Determine the x and y coordinates to where the ball strikes the slope. Also, determine the speed at which the ball hits the ground.

12-104. The projectile is launched with a velocity v0. Determine the range R, the maximum height h attained, and the time of flight. Express the results in terms of the angle 0 and y(). The acceleration due to gravity is g

Determine the horizontal velocity \(v_{A}\) of a tennis ball at A so that it just clears the net at B. Also, find the distance s where the ball strikes the ground.

12-106. The ball at A is kicked with a speed vA = 8 ft/s and at an angle 0A = 30. Determine the point (*,>') where it strikes the ground. Assume the ground has the shape of a parabola as shown.

The ball at A is kicked such that \(\theta_A=30^{\circ}\). If it strikes the ground at B having coordinates x = 15 ft, y = 9 ft, determine the speed at which it is kicked and the speed at which it strikes the ground.

The man at A wishes to throw two darts at the target at B so that they arrive at the same time. If each dart is thrown with a speed of 10 m/s, determine the angles \(\theta_{C}\) and \(\theta_{D}\) at which they should be thrown and the time between each throw. Note that the first dart must be thrown at \(\theta_{C}\left(>\theta_{D}\right)\), then the second dart is thrown at \(\theta_{D}\).

12-109. A boy throws a ball at O in the air with a speed v0 at an angle 0\. If he then throws another ball with the same speed Vo at an angle O2 < 0\, determine the time between the throws so that the bails collide in midair at B.

12-110. Small packages traveling on the conveyor belt fall off into a 1-m-long loading car. If the conveyor is running at a constant speed of vc = 2 in/s. determine the smallest and largest distance R at which the end A of the car may be placed from the conveyor so that the packages enter the car.

The fireman wishes to direct the flow of water from his hose to the fire at B. Determine two possible angles \(\theta_{1} \text { and } \theta_{2}\) at which this can be done. Water flows from the hose at \(v_{A}=80 \mathrm{\ ft} / \mathrm{s}\).

12-112. The baseball player A hits the baseball at vA = 40ft/s and 0A = 60 from the horizontal. When the ball is directly overhead of player B he begins to run under it. Determine the constant speed at which B must run and the distance d in order to make the catch at the same elevation at which the ball was hit.

12-113. The man stands 60 ft from the wall and throws a ball at it with a speed u0 = 50 ft/s. Determine the angle 0 at which he should release the ball so that it strikes the wall at the highest point possible. What is this height? The room has a ceiling height of 20 ft.

12-114. A car is traveling along a circular curve that has a radius of 50 m. If its speed is 16 m/s and is increasing uniformly at 8in/s2, determine the magnitude of its acceleration at this instant.

12-115. Determine the maximum constant speed a race car can have if the acceleration of the car cannot exceed 7.5 m/'s2 while rounding a track having a radius of curvature of 200 m.

A car moves along a circular track of radius \(250 \mathrm{ft}\) such that its speed for a short period of time \(0 \leq t \leq 4 \mathrm{~s}\), is \(v=3\left(t+t^2\right) \mathrm{ft} / \mathrm{s}\), where t is in seconds. Determine the magnitude of its acceleration when \(t=3 \mathrm{~s}\). How far has it traveled in \(t=3 \mathrm{~s}\)?

12-117. A car travels along a horizontal circular curved road that has a radius of 600 m. If the speed is uniformly increased at a rate of 2000 km/h2, determine the magnitude of the acceleration at the instant the speed of the car is 60 km/h.

The truck travels in a circular path having a radius of 50 m at a speed of v = 4 m/s. For a short distance from s = 0, its speed is increased by \(\dot{v}=(0.05\mathrm{\ s})\mathrm{\ m}/\mathrm{s}^2\), where s is in meters. Determine its speed and the magnitude of its acceleration when it has moved s =10 m.

The automobile is originally at rest at s=0. If its speed is increased by \(\dot{v}=\left(0.05 t^2\right) \mathrm{ft} / \mathrm{s}^2\), where t is in seconds, determine the magnitudes of its velocity and acceleration when \(t=18 \mathrm{~s}\).

*12-120. The automobile is originally at rest s = 0. If it then starts to increase its speed at v = (0.05/2) fl/s2, where / is in seconds, determine the magnitudes of its velocity and acceleration at 5 = 550 ft.

12-121. When the roller coaster is at B, it has a speed of 25 m/s, which is increasing at a, = 3 m/s2. Determine the magnitude of the acceleration of the roller coaster at this instant and the direction angle it makes with the x axis.

12-122. If the roller coaster starts from rest at A and its speed increases at a, = (6 - 0.065) m/s2, determine the magnitude of its acceleration when it reaches B where sg = 40 m.

12-123. The speedboat travels at a constant speed of 15 m/s while making a turn on a circular curve from A to B. If it takes 45 s to make the turn, determine the magnitude of the boats acceleration during the turn.

*12-124. The car travels along the circular path such that its speed is increased by a, = (0.5t/) m/s2, where t is in seconds. Determine the magnitudes of its velocity and acceleration after the car has traveled s = 18 m starting from rest. Neglect the size of the car.

12-125. The car passes point A with a speed of 25 m/s after which its speed is defined by v = (25 - 0.15s) m/s. Determine the magnitude of the cars acceleration when it reaches point B, where s = 51.5 m and .t = 50 m.

12-126. If the car passes point A with a speed of 20 m/s and begins to increase its speed at a constant rate of a, = 0.5 m/s2. determine the magnitude of the cars acceleration when s = 100 m and x = 0 .

12-127. A train is traveling with a constant speed of 14 m/s along the curved path. Determine the magnitude of the acceleration of the front of the train, B, at the instant it reaches point A (y = 0).

*12-128. When a car starts to round a curved road with the radius of curvature of 600 ft, it is traveling at 75 ft/s. If the cars speed begins to decrease at a rate of v = (-0.06/2) ft/s2, determine the magnitude of the acceleration of the car when it has traveled a distance of 5 = 700 ft.

12-129. When the motorcyclist is at A, he increases his speed along the vertical circular path at the rate of v = (0.3/) ft/s2, where / is in seconds. If he starts from rest at A, determine the magnitudes of his velocity and acceleration when he reaches B.

12-130. When the motorcyclist is at A. he increases his speed along the vertical circular path at the rate of v = (0.04s) ft/s2 where s is in ft. If he starts at vA = 2 ft/s where s = 0 at A, determine the magnitude of his velocity when he reaches B. Also, what is his initial acceleration?

12-131. At a given instant the train engine at E has a speed of 20 m/s and an acceleration of 14 m/s2 acting in the direction shown. Determine the rate of increase in the trains speed and the radius of curvature p of the path.

*12-132. Car B turns such that its speed is increased by (a,)[f = (0.5e') m/s2, where / is in seconds. If the car starts from rest when 6 = 0, determine the magnitudes of its velocity and acceleration when the arm A B rotates f) = 30. Neglect the size of the car.

12-133. Car B turns such that its speed is increased by (ar)H = (0.5e') m/s2, where / is in seconds. If the car starts from rest when 6 = 0, determine the magnitudes of its velocity and acceleration when / = 2 s. Neglect the size of the car.

12-134. A boat is traveling along a circular curve having a radius of 100 ft. If its speed at / = 0 is 15 ft/s and is increasing at v = (0.8/) ft/s2, determine the magnitude of its acceleration at the instant / = 5 s.

12-135. A boat is traveling along a circular path having a radius of 20 m. Determine the magnitude of the boats acceleration when the speed is v = 5 m/s and the rate of increase in the speed is v = 2 m/s2.

*12-136. Starting from rest, a bicyclist travels around a horizontal circular path, p = 10 m, at a speed of v = (0.09/2 + 0.1/) m/s, where t is in seconds. Determine the magnitudes of his velocity and acceleration when he has traveled s = 3 m.

A particle travels around a circular path having a radius of 50 m. If it is initially traveling with a speed of 10 m/s and its speed then increases at a rate of \(\dot{v}=(0.05 v) \mathrm{m} / \mathrm{s}^{2}\), determine the magnitude of the particle's acceleration four seconds later.

12-138. When (he bicycle passes point A, it has a speed of 6 m/s, which is increasing at the rate of v = 0.5 m/s2. Determine the magnitude of its acceleration when it is at point A.

12-139. The motorcycle is traveling at a constant speed of 60 km/h. Determine the magnitude of its acceleration when it is at point A.

*12-140. The jet plane travels along the vertical parabolic path. When it is at point A it has a speed of 200 m/s, which is increasing at the rate of 0.8 m/s2. Determine the magnitude of acceleration of the plane when it is at point A.

12-141. The ball is ejected horizontally from the tube with a speed of 8 m/s. Find the equation of the path, y = and then find the balls velocity and the normal and tangential components of acceleration when / = 0.25 s.

12-142. A toboggan is traveling down along a curve which can be approximated by the parabola y = 0.0 l.r2. Determine the magnitude of its acceleration when it reaches point A. where its speed is vA = 10 m/s, and it is increasing at the rate of vA = 3 m/s2.

12-143. A particle P moves along the curve y = (x2 4) m with a constant speed of 5 m/s. Determine the point on the curve where the maximum magnitude of acceleration occurs and compute its value.

*12-144. The Ferris wheel turns such that the speed of the passengers is increased by v = (4f) ft/s2, where t is in seconds. If the wheel starts from rest when 6 = 0, determine the magnitudes of the velocity and acceleration of the passengers when the wheel turns 0 = 30.

12-145. If the speed of the crate at A is 15 ft/s, which is increasing at a rate b = 3 ft/s2, determine the magnitude of the acceleration of the crate at this instant.

12-146. The race car has an initial speed vA = 15 m/s at A. If it increases its speed along the circular track at the rate a, = (0.4s) m/s2, where s is in meters, determine the time needed for the car to travel 20 m.Take p = 150 m.

12-147. A boy sits on a merry-go-round so that he is always located at r = 8 ft from the center of rotation. The merry-go-round is originally at rest.and then due to rotation the boy's speed is increased at 2 ft/s2. Determine the time needed for his acceleration to become 4 ft/s2.

*12-148. A particle travels along the path y = a + bx + c.r2, where a. b, c are constants. If the speed of the particle is constant, v = % determine the x and y components of velocity and the normal component of acceleration when .v 0.

The two particles A and B start at the origin O and travel in opposite directions along the circular path at constant speeds \(v_A=0.7\mathrm{\ m}/\mathrm{s}\) and \(v_B=1.5\mathrm{\ m}/\mathrm{s}\), respectively. Determine in t = 2 s, (a) the displacement along the path of each particle, (b) the position vector to each particle, and (c) the shortest distance between the particles.

12-150. The two particles A and B start at the origin O and travel in opposite directions along the circular path at constant speeds vA = 0.7 m/s and vR = 1.5 m/s, respectively. Determine the time when they collide and the magnitude of the acceleration of B just before this happens.

12-151. The position of a particle traveling along a curved path is s = (3t3 412 4- 4) m, where t is in seconds. When t = 2 s. the particle is at a position on the path where the radius of curvature is 25 m. Determine the magnitude of the particles acceleration at this instant.

*12-152. If the speed of the box at point A on the track is 30 ft/s which is increasing at the rate of v=5 ft/s2,determine the magnitude of the acceleration of the box at this instant.

12-153. A go-cart moves along a circular track of radius 100 ft such that its speed for a short period of time, 0 ^ / s 4s, is v = 60(1 e~l ) ft/s. Determine the magnitude of its acceleration when t = 2 s. How far has it traveled in / = 2 s? Use a numerical method to evaluate the integral.

12-154. The ball is kicked with an initial speed vA = 8 m/s at an angle 0A = 40 with the horizontal. Find the equation of the path, y = fix), and then determine the balls velocity and the normal and tangential components of its acceleration when t = 0.25 s.

12-155. The race car travels around the circular track with a speed of 16 m/s. When it reaches point A it increases its speed at a, = (f ;ly4) m/s2. where v is in m/s. Determine the magnitudes of the velocity and acceleration of the car when it reaches point B. Also, how much time is required for it to travel from A to 0?

*12-156. A particle P travels along an elliptical spiral path such that its position vector r is defined by r = {2 cos(0.1 /)i + 1.5 sin(0.1/)j + (2/)k} m, where t is in seconds and the arguments for the sine and cosine are given in radians. When t = 8 s. determine the coordinate direction angles a. /3. and y, which the binormal axis to the osculating plane makes with the x, y, and z axes. Hint: Solve for the velocity xP and acceleration aP of the particle in terms of their i. j, k components. The binormal is parallel to v> x aP. Why?

12-157. The motion of a particle is defined by the equations x = (21 + r) m and y = (r) m, where t is in seconds. Determine the normal and tangential components of the particles velocity and acceleration when t = 2 s.

12-158. The motorcycle travels along the elliptical track at a constant speed v. Determine the greatest magnitude of the acceleration if a > b.

12-159. A particle is moving along a circular path having a radius of 4 in. such that its position as a function of time is given by 0 cos 2r, where 0 is in radians and t is in seconds. Determine the magnitude of the acceleration of the particle when 0 = 30.

*12-160. A particle travels around a limaqon, defined by the equation r = b a cos 0, where a and b are constants. Determine the particles radial and transverse components of velocity and acceleration as a function of 0 and its time derivatives.

12-161. If a particles position is described by the polar coordinates /- = 4(1 + sin t) m and 0 = (2e~) rad, where t is in seconds and the argument for the sine is in radians, determine the radial and tangential components of the particles velocity and acceleration when / = 2 s.

12-162. An airplane is flying in a straight line with a velocity of 200 mi/h and an acceleration of 3 nii/h2. If the propeller has a diameter of 6 ft and is rotating at a constant angular rate of 120 rad/s. determine the magnitudes of velocity and acceleration of a particle located on the tip of the propeller.

12-163. A car is traveling along the circular curve of radius r = 300 ft. At the instant shown, its angular rate of rotation is 0 = 0.4 rad/s, which is increasing at the rate of 0 0.2 rad/s2. Determine the magnitudes of the cars velocity and acceleration at this instant.

*12-164. A radar gun at O rotates with the angular velocity of 0 =0.1 rad/s and angular acceleration of 0 = 0.025 rad/s2, at the instant 0 = 45, as it follows the motion of the car traveling along the circular road having a radius of r = 200 m. Determine the magnitudes of velocity and acceleration of the car at this instant.

12-165. If a particle moves along a path such that r = (2 cos t) ft and 0 = (//2) rad, where t is in seconds, plot the path r = f{0) and determine the particles radial and transverse components of velocity and acceleration.

12-166. If a particles position is described by the polar coordinates r = (2 sin 20) m and 0 = (40 rad, where l is in seconds, determine the radial and tangential components of its velocity and acceleration when / = 1 s.

12-167. The car travels along the circular curve having a radius r = 400 ft. At the instant shown, its angular rate of rotation is 0 = 0.025 rad/s. which is decreasing at the rate 0 = -0.008 rad/s2. Determine the radial and transverse components of the cars velocity and acceleration at this instant and sketch these components on the curve.

*12-168. The car travels along the circular curve of radius r = 400 ft with a constant speed of v = 30 ft/s. Determine the angular rate of rotation 0 of the radial line r and the magnitude of the cars acceleration.

12-169. The time rate of change of acceleration is referred to as the jerk, which is often used as a means of measuring passenger discomfort. Calculate this vector, a. in terms of its cylindrical components, using Eq. 12-32.

*12-170. A particle is moving along a circular path having a radius of 6 in. such that its position as a function of time is given by 0 = sin 3/, where 0 and the argument for the sine are in radians, and t is in seconds. Determine the acceleration of the particle at 0 = 30. The particle starts from rest at 0 = 0.

12-171. The slotted link is pinned at O, and as a result of the constant angular velocity 0 = 3 rad/s it drives the peg P for a short distance along the spiral guide r (0.4 0) m, where 0 is in radians. Determine the radial and transverse components of the velocity and acceleration of P at the instant 0 = 7r/3 rad.

12-172. Solve Prob. 12-171 if the slotted link has an angular acceleration 0 = 8 rad/s2 when 0 = 3 rad/s at 0 = 7t/3 rad.

12-173. The slotted link is pinned at O, and as a result of the constant angular velocity 0 = 3 rad/s it drives the peg P for a short distance along the spiral guide r (0.4 0) m, where 0 is in radians. Determine the velocity and acceleration of the particle at the instant it leaves the slot in the link. i.c.. when r = 0.5 m.

A particle moves in the x-y plane such that its position is defined by \(r=\left\{2 t \mathbf{i}+4 t^2 \mathbf{j}\right\} \mathrm{ft}\), where t is in seconds. Determine the radial and transverse components of the particle's velocity and acceleration when \(t=2 \mathrm{~s}\).

A particle P moves along the spiral path \(r=(10 / \theta) \mathrm{ft}\), where \(\theta\) is in radians. If it maintains a constant speed of \(v=20\ \mathrm{ft}/\mathrm{s}\), determine \(v_{r} \text { and } v_{\theta}\) as functions of \(\theta\) and evaluate each at \(\theta=1\mathrm{\ rad}\).

The driver of the car maintains a constant speed of \(40 \mathrm{~m} / \mathrm{s}\). Determine the angular velocity of the camera tracking the car when \(\theta=15^{\circ}\).

12-177. When 0 = 15, the car has a speed of 50m/s which is increasing at 6m/s2. Determine the angular velocity of the camera tracking the car at this instant.

12-178. The small washer slides down the cord OA. When it is at the midpoint, its speed is 200 mrn/s and its acceleration is 10 mm/s2. Express the velocity and acceleration of the washer at this point in terms of its cylindrical components.

12-179. A block moves outward along the slot in the platform with a speed of r (4t) m/s. where t is in seconds. The platform rotates at a constant rate of 6 rad/s. If the block starts from rest at the center, determine the magnitudes of its velocity and acceleration when t = 1 s.

*12-180. Pin P is constrained to move along the curve defined by the lemniscate r = (4 sin 20) ft. If the slotted arm OA rotates counterclockwise with a constant angular velocity of 0 = 1.5 rad/s, determine the magnitudes of the velocity and acceleration of peg P when 0 = 60.

12-181. Pin P is constrained to move along the curve defined by the lemniscate r = (4 sin 20) ft. If the angular position of the slotted arm OA is defined by 0 = rad. determine the magnitudes of the velocity and acceleration of the pin P when 0 = 60.

A cameraman standing at A is following the movement of a race car, B, which is traveling around a curved track at a constant speed of 30 m/s. Determine the angular rate \(\dot{\theta}\) at which the man must turn in order to keep the camera directed on the car at the instant \(\theta=30^{\circ} \text {. }\)

12-183. The slotted arm AB drives pin C through the spiral groove described by the equation r = aO. If the angular velocity is constant at 0, determine the radial and transverse components of velocity and acceleration of the pin.

*12-184. The slotted arm AB drives pin C through the spiral groove described by the equation r = (1.5 0) ft, where 0 is in radians. If the arm starts from rest when 0 = 60 and is driven at an angular velocity of 0 = (4r) rad/s, where t is in seconds, determine the radial and transverse components of velocity and acceleration of the pin C when / = 1 s.

12-185. If the slotted arm AB rotates counterclockwise with a constant angular velocity of 0 = 2 rad/s, determine the magnitudes of the velocity and acceleration of peg P at 0 30. The peg is constrained to move in the slots of the fixed bar CD and rotating bar A B.

12-186. The peg is constrained to move in the slots of the fixed bar CD and rotating bar A B. When 0 = 30, the angular velocity and angular acceleration of arm AB are 0 = 2 rad/s and 6=3 rad/s2, respectively. Determine the magnitudes of the velocity and acceleration of the peg P at this instant.

12-187. If the circular plate rotates clockwise with a constant angular velocity of 0 = 1.5 rad/s, determine the magnitudes of the velocity and acceleration of the follower rod AB when 0 = 2/3tt rad.

When \(\theta=2/3\ \pi\mathrm{\ rad}\), the angular velocity and angular acceleration of the circular plate are \(\dot{\theta}=1.5\mathrm{\ rad}/\mathrm{s}\) and \(\ddot{\theta}=3 \mathrm{\ rad} / \mathrm{s}^{2}\), respectively. Determine the magnitudes of the velocity and acceleration of the rod AB at this instant.

12-189. The box slides down the helical ramp with a constant speed of v = 2 m/s. Determine the magnitude of its acceleration. The ramp descends a vertical distance of 1 m for every full revolution.The mean radius of the ramp is r = 0.5 m.

12-190. The box slides down the helical ramp such that r = 0.5mj = (OJ/^rad, and z = (2-0.2r)m, where t is in seconds. Determine the magnitudes of the velocity and acceleration of the box at the instant 0 = 27rrad.

For a short distance the train travels along a track having the shape of a spiral, \(r=(1000 / \theta) \mathrm{m}\), where \(\theta\) is in radians. If it maintains a constant speed \(v=20 \mathrm{~m} / \mathrm{s}\), determine the radial and transverse components of its velocity when \(\theta=(9 \pi / 4) \mathrm{rad}\).

12-192. For a short distance the train travels along a track having the shape of a spiral, r = (1000/0) m. where 0 is in radians. If the angular rate is constant. 0 =0.2rad/s, determine the radial and transverse components of its velocity and acceleration when 0 = (9tt/4) rad.

A particle moves along an Archimedean spiral \(r=(8\theta)\mathrm{\ ft}\). where \(\theta\) is given in radians. If \(\dot{\theta}=4\mathrm{\ rad}/\mathrm{s}\) (constant), determine the radial and transverse components of the particle's velocity and acceleration at the instant \(\theta=\pi/2\mathrm{\ rad}\). Sketch the curve and show the components on the curve.

12-194. Solve Prob. 12-193 if the particle has an angular acceleration 0 = 5 rad/s2 when 0 = 4 rad/s at 0 = 7r/2 rad.

12-195. The arm of the robot has a fixed length of r = 3 ft and its grip A moves along the path z = O sin 40) ft, where 0 is in radians. If 0 = (0.5/) rad, where / is in seconds, determine the magnitudes of the grips velocity and acceleration when t = 3 s.

*12-196. For a short time the arm of the robot is extending at a constant rate such that r = 1.5 ft/s when r = 3 ft, z = (4r) ft, and 0 = 0.5/ rad, where t is in seconds. Determine the magnitudes of the velocity and acceleration of the grip A when / = 3 s.

The partial surface of the cam is that of a logarithmic spiral \(r=\left(40 e^{0.05 \theta}\right) \ mm\), where \(\theta\) is in radians. If the cam is rotating at a constant angular rate of \(\dot{\theta}=4 \mathrm{\ rad} / \mathrm{s}\), determine the magnitudes of the velocity and acceleration of the follower rod at the instant \(\theta=30^{\circ}\).

12-198. Solve Prob. 12-197, if the cam has an angular acceleration of 0 = 2 rad/s2 when its angular velocity is 0 = 4 rad/s at 0 = 30.

If the end of the cable at A is pulled down with a speed of 2 m/s, determine the speed at which block B rises.

*12-200. The motor at C pulls in the cable with an acceleration ac = (3r) m/s2, where t is in seconds. The motor at D draws in its cable at aD = 5 m/s2. If both motors start at the same instant from rest when d = 3 m. determine (a) the time needed from d = 0. and (b) the velocities of blocks A and B when this occurs.

The crate is being lifted up the inclined plane using the motor M and the rope and pulley arrangement shown. Determine the speed at which the cable must be taken up by the motor in order to move the crate up the plane with a constant speed of 4 ft/s.

12-202. Determine the time needed for the load at B to attain a speed of 8 m/s. starting from rest, if the cable is drawn into the motor with an acceleration of 0.2 m/s2.

Determine the displacement of the log if the truck at C pulls the cable 4 ft to the right.

*12-204. Determine the speed of cylinder A. if the rope is drawn toward the motor M at a constant rate of 10 m/s.

12-205. If the rope is drawn toward the motor M at a speed of vM = (5t'~) m/s. where t is in seconds, determine the speed of cylinder A when / = I s. 122(W*. If the hydraulic cylinder H draws in rod BC at 2 ft/s, determine the speed of slider A.

12-206 If the hydraulic cylinder H draws in rod BC at 2 ft/s, determine the speed of slider A.

If block A is moving downward with a speed of 4 ft/s while C is moving up at 2 ft/s, determine the velocity of block B.

If block A is moving downward at 6 ft/s while block C is moving down at 18 ft/s, determine the speed of block B.

Determine the displacement of block B if A is pulled down \(4 \mathrm{ft}\).

12-210. The pulley arrangement shown is designed for hoisting materials. If BC remains fixed while the plunger P is pushed downward with a speed of 4 ft/s, determine the speed of the load at A.

12-211. Determine the speed of block A if the end of the rope is pulled down with a speed of 4 m/s.

*12-212. The cylinder C is being lifted using the cable and pulley system shown. If point A on the cable is being drawn toward the drum with a speed of 2 m/s, determine the velocity of the cylinder.

12-213. The man pulls the boy up to the tree limb C by walking backward at a constant speed of 1.5 m/s. Determine the speed at which the boy is being lifted at the instant .v,, = 4 m. Neglect the size of the limb. When xA = 0, yH = 8 m, so that A and B are coincident, i.e., the rope is 16 m long.

12-214. The man pulls the boy up to the tree limb C by walking backward. If he starts from rest when .r4 = 0 and moves backward with a constant acceleration aA = 0.2 m/s2, determine the speed of the boy at the instant yH = 4 m. Neglect the size of the limb. When xA = 0. yB = 8 m, so that A and B are coincident, i.e.. the rope is 16 m long.

The roller at A is moving upward with a velocity of \(v_A=3 \mathrm{ft} / \mathrm{s}\) and has an acceleration of \(a_A=4 \mathrm{ft} / \mathrm{s}^2\) when \(s_A=4 \mathrm{ft}\). Determine the velocity and acceleration of block B at this instant.

The girl at C stands near the edge of the pier and pulls in the rope horizontally at a constant speed of \(6 \mathrm{ft} / \mathrm{s}\). Determine how fast the boat approaches the pier at the instant the rope length AB is \(50 \mathrm{ft}\).

12-217. The crate C is being lifted by moving the roller at A downward with a constant speed of vA = 2 m/s along the guide. Determine the velocity and acceleration of the crate at the instant s = I m. When the roller is at B. the crate rests on the ground. Neglect the size of the pulley in the calculation. Hint: Relate the coordinates .rc and xA using the problem geometry, then take the first and second time derivatives.

12-218. The man can row the boat in still water with a speed of 5 m/s. If the river is flowing at 2 m/s, determine the speed of the boat and the angle 0 he must direct the boat so that it travels from A to B.

12-219. Vertical motion of the load is produced by movement of the piston at A on the boom. Determine the distance the piston or pulley at C must move to the left in order to lift the load 2 ft. The cable is attached at B, passes over the pulley at C. then D, E, F, and again around E. and is attached at G.

12-220 If block B is moving down with a velocity i> and has an acceleration aH, determine the velocity and acceleration of block A in terms of the parameters shown.

Collars A and B are connected to the cord that passes over the small pulley at C. When A is located at D, B is 24 ft to the left of D. If A moves at a constant speed of 2 ft/s to the right, determine the speed of B when A is 4 ft to the right of D.

12-222. Two planes, A and B, are flying at the same vH = 500 km/h such that the angle between their straight-line courses is f) = 75, determine the velocity of plane B with respect to plane A.

12-223. At the instant shown, cars A and B are traveling at speeds of 55 mi/h and 40 mi/h, respectively. If B is increasing its speed by 1200 mi/h2, while A maintains a constant speed, determine the velocity and acceleration of B with respect to A. Car B moves along a curve having a radius of curvature of 0.5 mi. vA = 55 mi/h

*12-224. At the instant shown, car A travels along the straight portion of the road with a speed of 25 m/s. At this same instant car B travels along the circular portion of the road with a speed of 15 m/s. Determine the velocity of car B relative to car A.

12-225. An aircraft carrier is traveling forward with a has just taken off and has attained a forward horizontal air speed of 200km/h, measured from still water. If the plane at B is traveling along the runway of the carrier at 175 km/h in the direction shown, determine the velocity of A with respect to B. 50 km/h

12-226. A car is traveling north along a straight road at 50 km/h. An instrument in the car indicates that the wind is coming from the east. If the cars speed is 80 km/h, the instrument indicates that the wind is coming from the north-east. Determine the speed and direction of the wind.

12-227. Two boats leave the shore at the same time and travel in the directions shown. If vA = 20ft/s and Vp = 15 ft/s, determine the velocity of boat A with respect to boat B. How long after leaving the shore will the boats be 800 ft apart?

At the instant shown, the bicyclist at A is traveling at 7 m/s around the curve on the race track while increasing his speed at \(0.5 \mathrm{\ m} / \mathrm{s}^{2}\). The bicyclist at B is traveling at 8.5 m/s along the straight-a-way and increasing his speed at \(0.7 \mathrm{\ m} / \mathrm{s}^{2}\). Determine the relative velocity and relative acceleration of A with respect to B at this instant.

12-229. Cars A and B are traveling around the circular race track. At the instant shown. A has a speed of 90 ft/s and is increasing its speed at the rate of 15 ft/s2, whereas B has a speed of 105 ft/s and is decreasing its speed at 25 ft/s2. Determine the relative velocity and relative acceleration of car A with respect to car B at this instant.

12-230. The (wo cyclists A and B travel at the same constant speed v. Determine the speed of A with respect to B if A travels along the circular track, while B travels along the diameter of the circle.

12-233. A passenger in an automobile observes that raindrops make an angle of 30 with the horizontal as the auto travels forward with a speed of 60 km/h. Compute the terminal (constant) velocity v,. of the rain if it is assumed to fall vertically.

12-231. At the instant shown, cars A and B travel at speeds of 70 mi/h and 50 mi/h, respectively. If B is increasing its speed by 1100 mi/h2, while A maintains a constant speed, determine the velocity and acceleration of B with respect to A. Car B moves along a curve having a radius of curvature of 0.7 mi.

*12-232. At the instant shown, cars A and B travel at speeds of 70 mi/h and 50 mi/h, respectively. If B is decreasing its speed at 1400 mi/h2 while A is increasing its speed at 800 mi/h2, determine the acceleration of B with respect to A. Car B moves along a curve having a radius of curvature of 0.7 mi.

12-234. A man can swim at 4 ft/s in still water. He wishes to cross the 40-ft-wide river to point B, 30 ft downstream. If the river flows with a velocity of 2 ft/s. determine the speed of the man and the time needed to make the crossing. Note: While in the water he must not direct himself toward point B to reach this point. Why?

12-235. The ship travels at a constant speed of vs = 20m/s and the wind is blowing at a speed of vw = lOm/s, as shown. Determine the magnitude and direction of the horizontal component of velocity of the smoke coming from the smoke stack as it appears to a passenger on the ship.

*12-236. Car A travels along a straight road at a speed of 25 m/s while accelerating at 1.5 m/s2. At this same instant car C is traveling along the straight road with a speed of 30 m/s while decelerating at 3 m/s2. Determine the velocity and acceleration of car A relative to car C.

12-237. Car B is traveling along the curved road with a speed of 15 m/s w'hile decreasing its speed at 2 m/s2. At this same instant car C is traveling along the straight road with a speed of 30 m/s while decelerating at 3 m/s2. Determine the velocity and acceleration of car B relative to car C.

12-238. At a given instant the football player at A throws a football C with a velocity of 20 m/s in the direction shown. Determine the constant speed at which the player at B must run so that he can catch the football at the same elevation at which it was thrown. Also calculate the relative velocity and relative acceleration of the football with respect to B at the instant the catch is made. Player B is 15 m away from A when A starts to throw the football.

12-239. Both boats A and B leave the shore at O at the same time. If A travels at vA and B travels at i>B, write a general expression to determine the velocity of A with respect to B.

P12-1. If you measured the time it takes for the construction elevator to go from A to B. then B to C, and then C to D. and you also know the distance between each of the points, how could you determine the average velocity and average acceleration of the elevator as it ascends from A to D? Use numerical values to explain how this can be done.

P12-2. If the sprinkler at A is I m from the ground, then scale the necessary measurements from the photo to determine the approximate velocity of the water jet as it flows from the nozzle of the sprinkler.

The basketball was thrown at an angle measured from the horizontal to the man's outstretched arm. If the basket is 3 m from the ground, make appropriate measurements in the photo and determine if the ball located as shown will pass through the basket.

PI2-4. The pilot tells you the wingspan of her plane and her constant airspeed. How would you determine the acceleration of the plane at the moment shown? Use numerical values and take any necessary measurements from the photo.