The velocities of commuter trains A and B are as shown.

The motion of a particle is defined by the relation \(x=t^{4}-10 t^{2}+8t+12\), where x and t are expressed in inches and seconds, respectively. Determine the position, the velocity, and the acceleration of the particle when t = 1 s.

The motion of a particle is defined by the relation \(x=t^{3}-9 t^{2}+12t+10\), where x and t are expressed in feet and seconds, respectively. Determine the time, the position, and the acceleration of the particle when v = 0.

The vertical motion of mass A is defined by the relation x = 10 sin 2t + 15 cos 2t + 100, where x and t are expressed in millimeters and seconds, respectively. Determine (a) the position, velocity, and acceleration of A when t = 1 s, (b) the maximum velocity and acceleration of A.

A loaded railroad car is rolling at a constant velocity when it couples with a spring and dashpot bumper system. After the coupling, the motion of the car is defined by the relation \(x=60 e^{-4.5 t} \sin 16 t\), where x and t are expressed in millimeters and seconds, respectively. Determine the position, the velocity, and the acceleration of the railroad car when (a) t = 0, (b) t = 0.3 s.

The motion of a particle is defined by the relation \(x=6 t^{4}-2 t^{3}-12 t^{2}+3 t+3\), where x and t are expressed in meters and seconds, respectively. Determine the time, the position, and the velocity when a = 0.

The motion of a particle is defined by the relation \(x=t^{3}-9 t^{2}+24t-8\), where x and t are expressed in inches and seconds, respectively. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

The motion of a particle is defined by the relation \(x=2 t^{3}-15 t^{2}+24t+4\), where x is expressed in meters and t in seconds. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

The motion of a particle is defined by the relation \(x=t^{3}-6 t^{2}-36t-40\), where x and t are expressed in feet and seconds, respectively. Determine (a) when the velocity is zero, (b) the velocity, the acceleration, and the total distance traveled when x = 0.

The brakes of a car are applied, causing it to slow down at a rate of \(10\mathrm{\ ft}/\mathrm{s}^2\). Knowing that the car stops in 300 ft, determine (a) how fast the car was traveling immediately before the brakes were applied, (b) the time required for the car to stop.

The acceleration of a particle is directly proportional to the time t. At t = 0, the velocity of the particle is v = 16 in./s. Knowing that v = 15 in./s and that x = 20 in. when t = 1 s, determine the velocity, the position, and the total distance traveled when t = 7 s.

The acceleration of a particle is directly proportional to the square of the time t. When t = 0, the particle is at x = 24 m. Knowing that at t = 6 s, x = 96 m and v = 18 m/s, express x and v in terms of t.

The acceleration of a particle is defined by the relation \(a=k t^{2}\). (a) Knowing that v = -8 m/s when t = 0 and that v = +8 m/s when t = 2 s, determine the constant k. (b) Write the equations of motion, knowing also that x = 0 when t = 2 s.

The acceleration of point A is defined by the relation a = -1.8 sin kt, where a and t are expressed in \(\mathrm{m} / \mathrm{s}^{2}\) and seconds, respectively, and k = 3 rad/s. Knowing that x = 0 and v = 0.6 m/s when t = 0, determine the velocity and position of point A when t = 0.5 s.

The acceleration of point A is defined by the relation a = -1.08 sin kt -1.44 cos kt, where a and t are expressed in \(\mathrm{m} / \mathrm{s}^{2}\) and seconds, respectively, and k = 3 rad/s. Knowing that x = 0.16 m and v = 0.36 m/s when t = 0, determine the velocity and position of point A when t = 0.5 s.

A piece of electronic equipment that is surrounded by packing material is dropped so that it hits the ground with a speed of 4 m/s. After contact the equipment experiences an acceleration of a = -kx, where k is a constant and x is the compression of the packing material. If the packing material experiences a maximum compression of 20 mm, determine the maximum acceleration of the equipment.

A projectile enters a resisting medium at x = 0 with an initial velocity \(\mathbf{v}_0=900\mathrm{\ ft}/\mathrm{s}\) and travels 4 in. before coming to rest. Assuming that the velocity of the projectile is defined by the relation \(v=v_{0}-k x\), where v is expressed in ft/s and x is in feet, determine (a) the initial acceleration of the projectile, (b) the time required for the projectile to penetrate 3.9 in. into the resisting medium.

The acceleration of a particle is defined by the relation a = -k/x. It has been experimentally determined that v = 15 ft/s when x = 0.6 ft and that v = 9 ft/s when x = 1.2 ft. Determine (a) the velocity of the particle when x = 1.5 ft, (b) the position of the particle at which its velocity is zero.

A brass (nonmagnetic) block A and a steel magnet B are in equilibrium in a brass tube under the magnetic repelling force of another steel magnet C located at a distance x = 0.004 m from B. The force is inversely proportional to the square of the distance between B and C. If block A is suddenly removed, the acceleration of block B is \(a=-9.81+k / x^{2}\), where a and x are expressed in \(\mathrm{m} / \mathrm{s}^{2}\) and meters, respectively, and \(k=4\times10^{-4}\mathrm{\ m}^3/\mathrm{s}^2\). Determine the maximum velocity and acceleration of B.

Based on experimental observations, the acceleration of a particle is defined by the relation a = -(0.1 + sin x/b), where a and x are expressed in \(\mathrm{m} / \mathrm{s}^{2}\) and meters, respectively. Knowing that b = 0.8 m and that v = 1 m/s when x = 0, determine (a) the velocity of the particle when x = -1 m, (b) the position where the velocity is maximum, (c) the maximum velocity.

A spring AB is attached to a support at A and to a collar. The unstretched length of the spring is l. Knowing that the collar is released from rest at \(x=x_{0}\) and has an acceleration defined by the relation \(a=-100\left(x-l x / 2 \overline{ l^{2}+x^{2}}\right)\), determine the velocity of the collar as it passes through point C.

The acceleration of a particle is defined by the relation a = -0.8v, where a is expressed in \(\mathrm{m} / \mathrm{s}^{2}\) and v in m/s. Knowing that at t = 0 the velocity is 1 m/s, determine (a) the distance the particle will travel before coming to rest, (b) the time required for the particle’s velocity to be reduced by 50 percent of its initial value.

Starting from x = 0 with no initial velocity, a particle is given an acceleration \(a=0.12 \overline{v^{2}+16}\), where a and v are expressed in \(\mathrm{ft} / \mathrm{s}^{2}\) and ft/s, respectively. Determine (a) the position of the particle when v = 3 ft/s, (b) the speed and acceleration of the particle when x = 4 ft.

A ball is dropped from a boat so that it strikes the surface of a lake with a speed of 16.5 ft/s. While in the water the ball experiences an acceleration of a = 10 - 0.8v, where a and v are expressed in \(\mathrm{ft} / \mathrm{s}^{2}\) and ft/s, respectively. Knowing the ball takes 3 s to reach the bottom of the lake, determine (a) the depth of the lake, (b) the speed of the ball when it hits the bottom of the lake.

The acceleration of a particle is defined by the relation \(a=-k 1 \bar{v}\), where k is a constant. Knowing that x = 0 and v = 81 m/s at t = 0 and that v = 36 m/s when x = 18 m, determine (a) the velocity of the particle when x = 20 m, (b) the time required for the particle to come to rest.

A particle is projected to the right from the position x = 0 with an initial velocity of 9 m/s. If the acceleration of the particle is defined by the relation \(a=-0.6 v^{3 / 2}\), where a and v are expressed in \(\mathrm{m} / \mathrm{s}^{2}\) and m/s, respectively, determine (a) the distance the particle will have traveled when its velocity is 4 m/s, (b) the time when v = 1 m/s, (c) the time required for the particle to travel 6 m.

The acceleration of a particle is defined by the relation a = 0.4(1 - kv), where k is a constant. Knowing that at t = 0 the particle starts from rest at x = 4 m and that when t = 15 s, v = 4 m/s, determine (a) the constant k, (b) the position of the particle when v = 6 m/s, (c) the maximum velocity of the particle.

Experimental data indicate that in a region downstream of a given louvered supply vent the velocity of the emitted air is defined by \(v=0.18 v_{0} / x\), where v and x are expressed in m/s and meters, respectively, and \(v_{0}\) is the initial discharge velocity of the air. For \(v_0=3.6\ \mathrm{m/s}\), determine (a) the acceleration of the air at x = 2 m, (b) the time required for the air to flow from x = 1 to x = 3 m.

Based on observations, the speed of a jogger can be approximated by the relation \(v=7.5(1-0.04 x)^{0.3}\), where v and x are expressed in mi/h and miles, respectively. Knowing that x = 0 at t = 0, determine (a) the distance the jogger has run when t = 1 h, (b) the jogger’s acceleration in \(\mathrm{ft} / \mathrm{s}^{2}\) at t = 0, (c) the time required for the jogger to run 6 mi.

The acceleration due to gravity at an altitude y above the surface of the earth can be expressed as \(a=\frac{-32.2}{\left[1+\left(y / 20.9 \times 10^{6}\right)\right]^{2}}\) where a and y are expressed in \(\mathrm{ft} / \mathrm{s}^{2}\) and feet, respectively. Using this expression, compute the height reached by a projectile fired vertically upward from the surface of the earth if its initial velocity is (a) 1800 ft/s, (b) 3000 ft/s, (c) 36,700 ft/s.

The acceleration due to gravity of a particle falling toward the earth is \(a=-g R^{2} / r^{2}\), where r is the distance from the center of the earth to the particle, R is the radius of the earth, and g is the acceleration due to gravity at the surface of the earth. If R = 3960 mi, calculate the escape velocity, that is, the minimum velocity with which a particle must be projected vertically upward from the surface of the earth if it is not to return to the earth. (Hint: v = 0 for \(r=\infty\).)

The velocity of a particle is \(v=v_{0}[1-\sin (\mathrm{p} t / T)]\). Knowing that the particle starts from the origin with an initial velocity \(v_{0}\), determine (a) its position and its acceleration at t = 3T, (b) its average velocity during the interval t = 0 to t = T.

The velocity of a slider is defined by the relation \(v=v^{\prime} \sin \left(\mathrm{V}_{n} t+\mathrm{f}\right)\). Denoting the velocity and the position of the slider at t = 0 by \(v_{0}\) and \(x_{0}\), respectively, and knowing that the maximum displacement of the slider is \(2 x_{0}\), show that (a) \(v^{\prime}=\left(v_0^2+x_0^2\mathrm{v}_n^2\right)/2x_0\mathrm{V}_n\), (b) the maximum value of the velocity occurs when \(x=x_{0}[3-\left.\left(v_{0} / x_{0} \mathrm{~V}_{n}\right)^{2}\right] / 2\).

A stone is thrown vertically upward from a point on a bridge located 40 m above the water. Knowing that it strikes the water 4 s after release, determine (a) the speed with which the stone was thrown upward, (b) the speed with which the stone strikes the water.

A motorist is traveling at 54 km/h when she observes that a traffic light 240 m ahead of her turns red. The traffic light is timed to stay red for 24 s. If the motorist wishes to pass the light without stopping just as it turns green again, determine (a) the required uniform deceleration of the car, (b) the speed of the car as it passes the light.

A motorist enters a freeway at 30 mi/h and accelerates uniformly to 60 mi/h. From the odometer in the car, the motorist knows that she traveled 550 ft while accelerating. Determine (a) the acceleration of the car, (b) the time required to reach 60 mi/h.

A group of students launches a model rocket in the vertical direction. Based on tracking data, they determine that the altitude of the rocket was 89.6 ft at the end of the powered portion of the flight and that the rocket landed 16 s later. Knowing that the descent parachute failed to deploy so that the rocket fell freely to the ground after reaching its maximum altitude and assuming that \(g=32.2\mathrm{\ ft}/\mathrm{s}^2\), determine (a) the speed \(v_{1}\) of the rocket at the end of powered flight, (b) the maximum altitude reached by the rocket.

A small package is released from rest at A and moves along the skate wheel conveyor ABCD. The package has a uniform acceleration of \(4.8\mathrm{\ m}/\mathrm{s}^2\) as it moves down sections AB and CD, and its velocity is constant between B and C. If the velocity of the package at D is 7.2 m/s, determine (a) the distance d between C and D, (b) the time required for the package to reach D.

A sprinter in a 100-m race accelerates uniformly for the first 35 m and then runs with constant velocity. If the sprinter’s time for the first 35 m is 5.4 s, determine (a) his acceleration, (b) his final velocity, (c) his time for the race.

As relay runner A enters the 20-m-long exchange zone with a speed of 12.9 m/s, he begins to slow down. He hands the baton to runner B 1.82 s later as they leave the exchange zone with the same velocity. Determine (a) the uniform acceleration of each of the runners, (b) when runner B should begin to run.

In a boat race, boat A is leading boat B by 50 m and both boats are traveling at a constant speed of 180 km/h. At t = 0, the boats accelerate at constant rates. Knowing that when B passes A, t = 8 s and \(v_A=225\mathrm{\ km}/\mathrm{h}\), determine (a) the acceleration of A, (b) the acceleration of B.

A police officer in a patrol car parked in a 45 mi/h speed zone observes a passing automobile traveling at a slow, constant speed. Believing that the driver of the automobile might be intoxicated, the officer starts his car, accelerates uniformly to 60 mi/h in 8 s, and, maintaining a constant velocity of 60 mi/h, overtakes the motorist 42 s after the automobile passed him. Knowing that 18 s elapsed before the officer began pursuing the motorist, determine (a) the distance the officer traveled before overtaking the motorist, (b) the motorist’s speed.

Automobiles A and B are traveling in adjacent highway lanes and at t = 0 have the positions and speeds shown. Knowing that automobile A has a constant acceleration of \(1.8\mathrm{\ ft}/\mathrm{s}^2\) and that B has a constant deceleration of \(1.2\mathrm{\ ft}/\mathrm{s}^2\), determine (a) when and where A will overtake B, (b) the speed of each automobile at that time.

Two automobiles A and B are approaching each other in adjacent highway lanes. At t = 0, A and B are 3200 ft apart, their speeds are \(v_A=65\mathrm{\ mi}/\mathrm{h}\) and \(v_B=40\mathrm{\ mi}/\mathrm{h}\), and they are at points P and Q, respectively. Knowing that A passes point Q 40 s after B was there and that B passes point P 42 s after A was there, determine (a) the uniform accelerations of A and B, (b) when the vehicles pass each other, (c) the speed of B at that time.

An elevator is moving upward at a constant speed of 4 m/s. A man standing 10 m above the top of the elevator throws a ball upward with a speed of 3 m/s. Determine (a) when the ball will hit the elevator, (b) where the ball will hit the elevator with respect to the location of the man.

Two rockets are launched at a fireworks display. Rocket A is launched with an initial velocity \(v_0=100\mathrm{\ m}/\mathrm{s}\) and rocket B is launched \(t_1\mathrm{\ s}\) later with the same initial velocity. The two rockets are timed to explode simultaneously at a height of 300 m as A is falling and B is rising. Assuming a constant acceleration \(g=9.81\mathrm{\ m}/\mathrm{s}^2\), determine (a) the time \(t_{1}\), (b) the velocity of B relative to A at the time of the explosion.

Car A is parked along the northbound lane of a highway, and car B is traveling in the southbound lane at a constant speed of 60 mi/h. At t = 0, A starts and accelerates at a constant rate \(a_{A}\), while at t = 5 s, B begins to slow down with a constant deceleration of magnitude \(a_{A} / 6\). Knowing that when the cars pass each other x = 294 ft and \(v_{A}=v_{B}\), determine (a) the acceleration \(a_{A}\), (b) when the vehicles pass each other, (c) the distance d between the vehicles at t = 0.

The elevator shown in the figure moves downward with a constant velocity of 4 m/s. Determine (a) the velocity of the cable C, (b) the velocity of the counterweight W, (c) the relative velocity of the cable C with respect to the elevator, (d) the relative velocity of the counterweight W with respect to the elevator.

The elevator shown starts from rest and moves upward with a constant acceleration. If the counterweight W moves through 30 ft in 5 s, determine (a) the acceleration of the elevator and the cable C, (b) the velocity of the elevator after 5 s.

Slider block A moves to the left with a constant velocity of 6 m/s. Determine (a) the velocity of block B, (b) the velocity of portion D of the cable, (c) the relative velocity of portion C of the cable with respect to portion D.

Block B starts from rest and moves downward with a constant acceleration. Knowing that after slider block A has moved 9 in. its velocity is 6 ft/s, determine (a) the accelerations of A and B, (b) the velocity and the change in position of B after 2 s.

Slider block B moves to the right with a constant velocity of 300 mm/s. Determine (a) the velocity of slider block A, (b) the velocity of portion C of the cable, (c) the velocity of portion D of the cable, (d) the relative velocity of portion C of the cable with respect to slider block A.

At the instant shown, slider block B is moving with a constant acceleration, and its speed is 150 mm/s. Knowing that after slider block A has moved 240 mm to the right its velocity is 60 mm/s, determine (a) the accelerations of A and B, (b) the acceleration of portion D of the cable, (c) the velocity and the change in position of slider block B after 4 s.

Collar A starts from rest and moves upward with a constant acceleration. Knowing that after 8 s the relative velocity of collar B with respect to collar A is 24 in./s, determine (a) the accelerations of A and B, (b) the velocity and the change in position of B after 6 s.

The motor M reels in the cable at a constant rate of 100 mm/s. Determine (a) the velocity of load L, (b) the velocity of pulley B with respect to load L.

Block C starts from rest at t = 0 and moves downward with a constant acceleration of \(4\mathrm{\ in}./\mathrm{s}^2\). Knowing that block B has a constant velocity of 3 in./s upward, determine (a) the time when the velocity of block A is zero, (b) the time when the velocity of block A is equal to the velocity of block D, (c) the change in position of block A after 5 s.

Block A starts from rest at t = 0 and moves downward with a constant acceleration of \(6\mathrm{\ in}./\mathrm{s}^2\). Knowing that block B moves up with a constant velocity of 3 in./s, determine (a) the time when the velocity of block C is zero, (b) the corresponding position of block C.

Block B starts from rest, block A moves with a constant accelera-tion, and slider block C moves to the right with a constant acceleration of \(75\mathrm{\ mm}/\mathrm{s}^2\). Knowing that at t = 2 s the velocities of B and C are 480 mm/s downward and 280 mm/s to the right, respectively, determine (a) the accelerations of A and B, (b) the initial velocities of A and C, (c) the change in position of slider block C after 3 s.

Block B moves downward with a constant velocity of 20 mm/s. At t = 0, block A is moving upward with a constant acceleration, and its velocity is 30 mm/s. Knowing that at t = 3 s slider block C has moved 57 mm to the right, determine (a) the velocity of slider block C at t = 0, (b) the accelerations of A and C, (c) the change in position of block A after 5 s.

The system shown starts from rest, and each component moves with a constant acceleration. If the relative acceleration of block C with respect to collar B is \(60\mathrm{\ mm}/\mathrm{s}^2\) upward and the relative acceleration of block D with respect to block A is \(110\mathrm{\ mm}/\mathrm{s}^2\) downward, determine (a) the velocity of block C after 3 s, (b) the change in position of block D after 5 s.

The system shown starts from rest, and the length of the upper cord is adjusted so that A, B, and C are initially at the same level. Each component moves with a constant acceleration, and after 2 s the relative change in position of block C with respect to block A is 280 mm upward. Knowing that when the relative velocity of collar B with respect to block A is 80 mm/s downward, the displacements of A and B are 160 mm downward and 320 mm downward, respectively, determine (a) the accelerations of A and B if \(a_B>10\mathrm{\ mm}/\mathrm{s}^2\), (b) the change in position of block D when the velocity of block C is 600 mm/s upward.

A particle moves in a straight line with the acceleration shown in the figure. Knowing that it starts from the origin with \(v_0=-14\mathrm{\ ft}/\mathrm{s}\), plot the v–t and x–t curves for 0 < t < 15 s and determine (a) the maximum value of the velocity of the particle, (b) the maximum value of its position coordinate.

For the particle and motion of Prob. 11.61, plot the v–t and x–t curves for 0 < t < 15 s and determine the velocity of the particle, its position, and the total distance traveled after 10 s.

A particle moves in a straight line with the velocity shown in the figure. Knowing that x = -540 m at t = 0, (a) construct the a–t and x–t curves for 0 < t < 50 s, and determine (b) the total distance traveled by the particle when t = 50 s, (c) the two times at which x = 0.

A particle moves in a straight line with the velocity shown in the figure. Knowing that x = -540 m at t = 0, (a) construct the a–t and x–t curves for 0 < t < 50 s, and determine (b) the maximum value of the position coordinate of the particle, (c) the values of t for which the particle is at x = 100 m.

During a finishing operation the bed of an industrial planer moves alternately 30 in. to the right and 30 in. to the left. The velocity of the bed is limited to a maximum value of 6 in./s to the right and 12 in./s to the left; the acceleration is successively equal to \(6\ \mathrm{in.}/\mathrm{s}^2\) to the right, zero, \(6\ \mathrm{in.}/\mathrm{s}^2\) to the left, zero, etc. Determine the time required for the bed to complete a full cycle, and draw the v–t and x–t curves.

A parachutist is in free fall at a rate of 200 km/h when he opens his parachute at an altitude of 600 m. Following a rapid and constant deceleration,he then descends at a constant rate of 50 km/h from 586 m to 30 m, where he maneuvers the parachute into the wind to further slow his descent. Knowing that the parachutist lands with a negligible downward velocity, determine (a) the time required for the parachutist to land after opening his parachute, (b) the initial deceleration.

A commuter train traveling at 40 mi/h is 3 mi from a station. The train then decelerates so that its speed is 20 mi/h when it is 0.5 mi from the station. Knowing that the train arrives at the station 7.5 min after beginning to decelerate and assuming constant decelerations, determine (a) the time required for the train to travel the first 2.5 mi, (b) the speed of the train as it arrives at the station, (c) the final constant deceleration of the train.

A temperature sensor is attached to slider AB which moves back and forth through 60 in. The maximum velocities of the slider are 12 in./s to the right and 30 in./s to the left. When the slider is moving to the right, it accelerates and decelerates at a constant rate of \(6\mathrm{\ in.}/\mathrm{s}^2\); when moving to the left, the slider accelerates and decelerates at a constant rate of \(20\mathrm{\ in.}/\mathrm{s}^2\). Determine the time required for the slider to complete a full cycle, and construct the v–t and x–t curves of its motion.

In a water-tank test involving the launching of a small model boat, the model’s initial horizontal velocity is 6 m/s and its horizontal acceleration varies linearly from \(-12\mathrm{\ m}/\mathrm{s}^2\) at t = 0 to \(-2\mathrm{\ m}/\mathrm{s}^2\) at \(t=t_{1}\) and then remains equal to \(-2\mathrm{\ m}/\mathrm{s}^2\) until t = 1.4 s. Know-ing that v = 1.8 m/s when \(t=t_{1}\), determine (a) the value of \(t_{1}\), (b) the velocity and the position of the model at t = 1.4 s.

The acceleration record shown was obtained for a small airplane traveling along a straight course. Knowing that x = 0 and v = 60 m/s when t = 0, determine (a) the velocity and position of the plane at t = 20 s, (b) its average velocity during the interval 6 s < t < 14 s.

In a 400-m race, runner A reaches her maximum velocity \(v_{A}\) in 4 s with constant acceleration and maintains that velocity until she reaches the halfway point with a split time of 25 s. Runner B reaches her maximum velocity \(v_{B}\) in 5 s with constant acceleration and maintains that velocity until she reaches the halfway point with a split time of 25.2 s. Both runners then run the second half of the race with the same constant deceleration of \(0.1\mathrm{\ m}/\mathrm{s}^2\). Determine (a) the race times for both runners, (b) the position of the winner relative to the loser when the winner reaches the finish line.

A car and a truck are both traveling at the constant speed of 35 mi/h; the car is 40 ft behind the truck. The driver of the car wants to pass the truck, i.e., he wishes to place his car at B, 40 ft in front of the truck, and then resume the speed of 35 mi/h. The maximum acceleration of the car is \(5\mathrm{\ ft}/\mathrm{s}^2\) and the maximum deceleration obtained by applying the brakesis \(20\mathrm{\ ft}/\mathrm{s}^2\). What is the shortest time in which the driver of the car can complete the passing operation if he does not at any time exceed a speed of 50 mi/h? Draw the v–t curve.

Solve Prob. 11.72, assuming that the driver of the car does not pay any attention to the speed limit while passing and concentrates on reaching position B and resuming a speed of 35 mi/h in the shortest possible time. What is the maximum speed reached? Draw the v–t curve.

Car A is traveling on a highway at a constant speed \(\left(v_A\right)_0=60\mathrm{\ mi}/\mathrm{h}\) and is 380 ft from the entrance of an access ramp when car B enters the acceleration lane at that point at a speed \(\left(v_B\right)_0=15\mathrm{\ mi}/\mathrm{h}\). Car B accelerates uniformly and enters the main traffic lane after traveling 200 ft in 5 s. It then continues to accelerate at the same rate until it reaches a speed of 60 mi/h, which it then maintains. Determine the final distance between the two cars.

An elevator starts from rest and moves upward, accelerating at a rate of \(1.2\mathrm{\ m}/\mathrm{s}^2\) until it reaches a speed of 7.8 m/s, which it then maintains. Two seconds after the elevator begins to move, a man standing 12 m above the initial position of the top of the elevator throws a ball upward with an initial velocity of 20 m/s. Determine when the ball will hit the elevator.

Car A is traveling at 40 mi/h when it enters a 30 mi/h speed zone. The driver of car A decelerates at a rate of \(16\mathrm{\ ft}/\mathrm{s}^2\) until reaching a speed of 30 mi/h, which she then maintains. When car B, which was initially 60 ft behind car A and traveling at a constant speed of 45 mi/h, enters the speed zone, its driver decelerates at a rate of \(20\mathrm{\ ft}/\mathrm{s}^2\) until reaching a speed of 28 mi/h. Knowing that the driver of car B maintains a speed of 28 mi/h, determine (a) the closest that car B comes to car A, (b) the time at which car A is 70 ft in front of car B.

An accelerometer record for the motion of a given part of a mechanism is approximated by an arc of a parabola for 0.2 s and a straight line for the next 0.2 s as shown in the figure. Knowing that v = 0 when t = 0 and x = 0.8 ft when t = 0.4 s, (a) construct the v – t curve for \(0\le t\le0.4\mathrm{\ s}\), (b) determine the position of the part at t = 0.3 s and t = 0.2 s.

A car is traveling at a constant speed of 54 km/h when its driver sees a child run into the road. The driver applies her brakes until the child returns to the sidewalk and then accelerates to resume her original speed of 54 km/h; the acceleration record of the car is shown in the figure. Assuming x = 0 when t = 0, determine (a) the time \(t_{1}\) at which the velocity is again 54 km/h, (b) the position of the car at that time, (c) the average velocity of the car during the interval \(1\mathrm{\ s}\le t\le t_1\).

An airport shuttle train travels between two terminals that are 1.6 mi apart. To maintain passenger comfort, the acceleration of the train is limited to \(\pm4\mathrm{\ ft}/\mathrm{s}^2\), and the jerk, or rate of change of acceleration, is limited to \(\pm0.8\mathrm{\ ft}/\mathrm{s}^2\) per second. If the shuttle has a maximum speed of 20 mi/h, determine (a) the shortest time for the shuttle to travel between the two terminals, (b) the corresponding average velocity of the shuttle.

During a manufacturing process, a conveyor belt starts from rest and travels a total of 1.2 ft before temporarily coming to rest. Knowing that the jerk, or rate of change of acceleration, is limited to \(\pm4.8\mathrm{\ ft}/\mathrm{s}^2\) per second, determine (a) the shortest time required for the belt to move 1.2 ft, (b) the maximum and average values of the velocity of the belt during that time.

Two seconds are required to bring the piston rod of an air cylinder to rest; the acceleration record of the piston rod during the 2 s is as shown. Determine by approximate means (a) the initial velocity of the piston rod, (b) the distance traveled by the piston rod as it is brought to rest.

The acceleration record shown was obtained during the speed trials of a sports car. Knowing that the car starts from rest, determine by approximate means (a) the velocity of the car at t = 8 s, (b) the distance the car has traveled at t = 20 s.

A training airplane has a velocity of 126 ft/s when it lands on an aircraft carrier. As the arresting gear of the carrier brings the airplane to rest, the velocity and the acceleration of the airplane are recorded; the results are shown (solid curve) in the figure. Determine by approximate means (a) the time required for the airplane to come to rest, (b) the distance traveled in that time.

Shown in the figure is a portion of the experimentally determined v–x curve for a shuttle cart. Determine by approximate means the acceleration of the cart when (a) x = 10 in., (b) v = 80 in./s.

Using the method of Sec. 11.8, derive the formula \(x=x_{0}+v_{0} t+\frac{1}{2} a t^{2}\) for the position coordinate of a particle in uniformly accelerated rectilinear motion.

Using the method of Sec. 11.8, determine the position of the particle of Prob. 11.61 when t = 8 s.

The acceleration of an object subjected to the pressure wave of a large explosion is defined approximately by the curve shown. The object is initially at rest and is again at rest at time \(t_1\). Using the method of Sec. 11.8, determine (a) the time \(t_1\), (b) the distance through which the object is moved by the pressure wave.

For the particle of Prob. 11.63, draw the a–t curve and determine, using the method of Sec. 11.8, (a) the position of the particle when t = 52 s, (b) the maximum value of its position coordinate.

A ball is thrown so that the motion is defined by the equations x = 5t and \(y = 2 + 6t - 4.9t^{2}\), where x and y are expressed in meters and t is expressed in seconds. Determine (a) the velocity at t = 1 s, (b) the horizontal distance the ball travels before hitting the ground.

The motion of a vibrating particle is defined by the position vector \(\mathbf{r}=10\left(1-e^{-3 t}\right) \mathbf{i}+\left(4 e^{-2 t} \sin 15 t\right) \mathbf{j}\), where r and t are expressed in millimeters and seconds, respectively. Determine the velocity and acceleration when (a) t = 0, (b) t = 0.5 s.

The motion of a vibrating particle is defined by the position vector \(\mathbf{r}=(4 \sin \mathrm{p} t) \mathbf{i}-(\cos 2 \mathrm{p} t) \mathbf{j}\), where r is expressed in inches and t in seconds. (a) Determine the velocity and acceleration when t = 1 s. (b) Show that the path of the particle is parabolic.

The motion of a particle is defined by the equations x = 10t - 5 sin t and y = 10 - 5 cos t, where x and y are expressed in feet and t is expressed in seconds. Sketch the path of the particle for the time interval \(0 \leq t \leq 2 \mathrm{p}\), and determine (a) the magnitudes of the smallest and largest velocities reached by the particle, (b) the corresponding times, positions, and directions of the velocities.

The damped motion of a vibrating particle is defined by the position vector \(\boldsymbol{r}=x_{1}[1-1 /(t+1)] \mathbf{i}+\left(y_{1} e^{-pt / 2} \cos 2 \mathrm{p} t\right) \mathbf{j}\), where t is expressed in seconds. For \(x_{1}=30 \ \mathrm{mm}\) and \(y_{1}=20 \ \mathrm{mm}\), determine the position, the velocity, and the acceleration of the particle when (a) t = 0, (b) t = 1.5 s.

The motion of a particle is defined by the position vector \(\boldsymbol{r}=A(\cos t+t \sin t) \mathbf{i}+A(\sin t-t \cos t) \mathbf{j}\), where t is expressed in seconds. Determine the values of t for which the position vector and the acceleration are (a) perpendicular, (b) parallel.

The three-dimensional motion of a particle is defined by the position vector \(\boldsymbol{r}=\left(R t \cos \mathrm{V}_{n} t\right) \mathbf{i}+c t \mathbf{j}+\left(R t \sin \mathrm{v}_{n} t\right) \mathbf{k}\). Determine the magnitudes of the velocity and acceleration of the particle. (The space curve described by the particle is a conic helix.)

The three-dimensional motion of a particle is defined by the position vector \(\boldsymbol{r}=(A t \cos t) \mathbf{i}+\left(A 2 \overline {t^{2}+1}\right) \mathbf{j}+(B t \sin t) \mathbf{k}\), where r and t are expressed in feet and seconds, respectively. Show that the curve described by the particle lies on the hyperboloid \((y / A)^{2}-(x / A)^{2}-(z / B)^{2}=1\). For A = 3 and B = 1, determine (a) the magnitudes of the velocity and acceleration when t = 0, (b) the smallest nonzero value of t for which the position vector and the velocity are perpendicular to each other.

An airplane used to drop water on brushfires is flying horizontally in a straight line at 180 mi/h at an altitude of 300 ft. Determine the distance d at which the pilot should release the water so that it will hit the fire at B.

A helicopter is flying with a constant horizontal velocity of 180 km/h and is directly above point A when a loose part begins to fall. The part lands 6.5 s later at point B on an inclined surface. Determine (a) the distance d between points A and B. (b) the initial height h.

A baseball pitching machine “throws” baseballs with a horizontal velocity \(v_{0}\). Knowing that height h varies between 788 mm and 1068 mm, determine (a) the range of values of \(v_{0}\), (b) the values of a corresponding to h = 788 mm and h = 1068 mm.

While delivering newspapers, a girl throws a newspaper with a horizontal velocity \(v_{0}\). Determine the range of values of \(v_{0}\) if the newspaper is to land between points B and C.

Water flows from a drain spout with an initial velocity of 2.5 ft/s at an angle of \(15^{\circ}\) with the horizontal. Determine the range of values of the distance d for which the water will enter the trough BC.

Milk is poured into a glass of height 140 mm and inside diameter Problems 66 mm. If the initial velocity of the milk is 1.2 m/s at an angle of \(40^{\circ}\) with the horizontal, determine the range of values of the height h for which the milk will enter the glass.

A volleyball player serves the ball with an initial velocity \(v_{0}\) of magnitude 13.40 m/s at an angle of \(20^{\circ}\) with the horizontal. Determine (a) if the ball will clear the top of the net, (b) how far from the net the ball will land.

A golfer hits a golf ball with an initial velocity of 160 ft/s at an angle of \(25^{\circ}\) with the horizontal. Knowing that the fairway slopes downward at an average angle of \(5^{\circ}\), determine the distance d between the golfer and point B where the ball first lands.

A homeowner uses a snowblower to clear his driveway. Knowing that the snow is discharged at an average angle of \(40^{\circ}\) with the horizontal, determine the initial velocity \(v_{0}\) of the snow.

At halftime of a football game souvenir balls are thrown to the spectators with a velocity \(v_{0}\). Determine the range of values of \(v_{0}\) if the balls are to land between points B and C.

A basketball player shoots when she is 16 ft from the backboard. Knowing that the ball has an initial velocity \(v_{0}\) at an angle of \(30^{\circ}\) with the horizontal, determine the value of \(v_{0}\) when d is equal to (a) 9 in., (b) 17 in.

A tennis player serves the ball at a height h = 2.5 m with an initial velocity of \(v_{0}\) at an angle of \(5^{\circ}\) with the horizontal. Determine the range of \(v_{0}\) for which the ball will land in the service area that extends to 6.4 m beyond the net.

The nozzle at A Problems discharges cooling water with an initial velocity \(v_{0}\) at an angle of \(6^{\circ}\) with the horizontal onto a grinding wheel 350 mm in diameter. Determine the range of values of the initial velocity for which the water will land on the grinding wheel between points B and C.

While holding one of its ends, a worker lobs a coil of rope over the lowest limb of a tree. If he throws the rope with an initial velocity \(\mathbf{v}_{0}\) at an angle of \(65^{\circ}\) with the horizontal, determine the range of values of \(v_{0}\) for which the rope will go over only the lowest limb.

The pitcher in a softball game throws a ball with an initial velocity \(v_{0}\) of 72 km/h at an angle a with the horizontal. If the height of the ball at point B is 0.68 m, determine (a) the angle a, (b) the angle u that the velocity of the ball at point B forms with the horizontal.

A model rocket is launched from point A with an initial velocity \(v_{0}\) of 75 m/s. If the rocket’s descent parachute does not deploy and the rocket lands a distance d = 100 m from A, determine (a) the angle a that \(v_{0}\) forms with the vertical, (b) the maximum height above point A reached by the rocket, (c) the duration of the flight.

The initial velocity \(v_{0}\) of a hockey puck is 105 mi/h. Determine (a) the largest value (less than \(45^{\circ}\)) of the angle a for which the puck will enter the net, (b) the corresponding time required for the puck to reach the net.

A worker uses high-pressure water to clean the inside of a long drainpipe. If the water is discharged with an initial velocity \(v_{0}\) of 11.5 m/s, determine (a) the distance d to the farthest point B on the top of the pipe that the worker can wash from his position at A, (b) the corresponding angle a.

An oscillating garden sprinkler which discharges water with an initial velocity \(v_{0}\) of 8 m/s is used to water a vegetable garden. Determine the distance d to the farthest point B that will be watered and the corresponding angle a when (a) the vegetables are just beginning to grow, (b) the height h of the corn is 1.8 m.

A mountain climber plans to jump from A to B over a crevasse. Determine the smallest value of the climber’s initial velocity \(v_{0}\) and the corresponding value of angle a so that he lands at B.

The velocities of skiers A and B are as shown. Determine the velocity of A with respect to B.

The three blocks shown move with constant velocities. Find the velocity of each block, knowing that the relative velocity of A with respect to C is 300 mm/s upward and that the relative velocity of B with respect to A is 200 mm/s downward.

Three seconds after automobile B passes through the intersection shown, automobile A passes through the same intersection. Knowing that the speed of each automobile is constant, determine (a) the relative velocity of B with respect to A, (b) the change in position of B with respect to A during a 4-s interval, (c) the distance between the two automobiles 2 s after A has passed through the intersection.

Shore-based radar indicates that a ferry leaves its slip with a velocity v = 18 km/h d \(70^{\circ}\), while instruments aboard the ferry indicate a speed of 18.4 km/h and a heading of \(30^{\circ}\) west of south relative to the river. Determine the velocity of the river.

Airplanes A and B are flying at the same altitude and are tracking the eye of hurricane C. The relative velocity of C with respect to A is \(\mathbf{v}_{C / A}=350 \ \mathrm{km} / \mathrm{h} \ \mathrm{d} \ 75^{\circ}\), and the relative velocity of C with respect to B is \(\mathbf{v}_{C / B}=400 \ \mathrm{km} / \mathrm{h} \ \mathrm{C} \ 40^{\circ}\). Determine (a) the relative velocity of B with respect to A, (b) the velocity of A if ground-based radar indicates that the hurricane is moving at a speed of 30 km/h due north, (c) the change in position of C with respect to B during a 15-min interval.

Pin P moves at a constant speed of 150 mm/s in a counterclockwise sense along a circular slot which has been milled in the slider block A shown. Knowing that the block moves downward at a constant speed of 100 mm/s, determine the velocity of pin P when (a) \(\mathrm{u}=30^{\circ}\), (b) \(\mathrm{u}=120^{\circ}\).

Knowing that at the instant shown assembly A has a velocity of 9 in./s and an acceleration of \(15 \text { in. } / \mathrm{s}^{2}\) both directed downward, determine (a) the velocity of block B, (b) the acceleration of block B.

Knowing that at the instant shown block A has a velocity of 8 in./s and an acceleration of \(6 \text { in. } / \mathrm{s}^{2}\) both directed down the incline, determine (a) the velocity of block B, (b) the acceleration of block B.

A boat is moving to the right with a constant deceleration of \(0.3 \ \mathrm{m} / \mathrm{s}^{2}\) when a boy standing on the deck D throws a ball with an initial velocity relative to the deck which is vertical. The ball rises to a maximum height of 8 m above the release point and the boy must step forward a distance d to catch it at the same height as the release point. Determine (a) the distance d, (b) the relative velocity of the ball with respect to the deck when the ball is caught.

The assembly of rod A and wedge B starts from rest and moves to the right with a constant acceleration of \(2 \ \mathrm{mm} / \mathrm{s}^{2}\). Determine (a) the acceleration of wedge C, (b) the velocity of wedge C when t = 10 s.

Determine the required velocity of the belt B if the relative velocity with which the sand hits belt B is to be (a) vertical, (b) as small as possible.

Conveyor belt A, which forms a \(20^{\circ}\) angle with the horizontal, moves at a constant speed of 4 ft/s and is used to load an airplane. Knowing that a worker tosses duffel bag B with an initial velocity of 2.5 ft/s at an angle of \(30^{\circ}\) with the horizontal, determine the velocity of the bag relative to the belt as it lands on the belt.

During a rainstorm the paths of the raindrops appear to form an angle of \(30^{\circ}\) with the vertical and to be directed to the left when observed from a side window of a train moving at a speed of 15 km/h. A short time later, after the speed of the train has increased to 24 km/h, the angle between the vertical and the paths of the drops appears to be \(45^{\circ}\). If the train were stopped, at what angle and with what velocity would the drops be observed to fall?

As observed from a ship moving due east at 9 km/h, the wind appears to blow from the south. After the ship has changed course and speed, and as it is moving north at 6 km/h, the wind appears to blow from the southwest. Assuming that the wind velocity is constant during the period of observation, determine the magnitude and direction of the true wind velocity.

When a small boat travels north at 5 km/h, a flag mounted on its stern forms an angle \(u = 50^{\circ}\) with the centerline of the boat as shown. A short time later, when the boat travels east at 20 km/h, angle u is again \(50^{\circ}\). Determine the speed and the direction of the wind.

As part of a department store display, a model train D runs on a slight incline between the store’s up and down escalators. When the train and shoppers pass point A, the train appears to a shopper on the up escalator B to move downward at an angle of \(22^{\circ}\) with the horizontal, and to a shopper on the down escalator C to move upward at an angle of \(23^{\circ}\) with the horizontal and to travel to the left. Knowing that the speed of the escalators is 3 ft/s, determine the speed and the direction of the train.

Determine the smallest radius that should be used for a highway if the normal component of the acceleration of a car traveling at 72 km/h is not to exceed \(0.8 \ m/s^{2}\).

Determine the maximum speed that the cars of the roller-coaster can reach along the circular portion AB of the track if \(\rho = 25 \ m\) and the normal component of their acceleration cannot exceed 3g.

A bull-roarer is a piece of wood that produces a roaring sound when Problems attached to the end of a string and whirled around in a circle. Determine the magnitude of the normal acceleration of a bull-roarer when it is spun in a circle of radius 0.9 m at a speed of 20 m/s.

To test its performance, an automobile is driven around a circular test track of diameter d. Determine (a) the value of d if when the speed of the automobile is 45 mi/h, the normal component of the acceleration is \(11 \ ft/s^{2}\), (b) the speed of the automobile if d = 600 ft and the normal component of the acceleration is measured to be 0.6g.

An outdoor track is 420 ft in diameter. A runner increases her speed at a constant rate from 14 to 24 ft/s over a distance of 95 ft. Determine the magnitude of the total acceleration of the runner 2 s after she begins to increase her speed.

A robot arm moves so that P travels in a circle about point B, which is not moving. Knowing that P starts from rest, and its speed increases at a constant rate of \(10 \ mm/s^{2}\), determine (a) the magnitude of the acceleration when t = 4 s, (b) the time for the magnitude of the acceleration to be \(80 \ mm/s^{2}\).

A monorail train starts from rest on a curve of radius 400 m and accelerates at the constant rate \(a_{t}\). If the maximum total acceleration of the train must not exceed \(1.5 \ m/s^{2}\), determine (a) the shortest distance in which the train can reach a speed of 72 km/h, (b) the corresponding constant rate of acceleration \(a_{t}\).

A motorist starts from rest at point A on a circular entrance ramp when t = 0, increases the speed of her automobile at a constant rate and enters the highway at point B. Knowing that her speed continues to increase at the same rate until it reaches 100 km/h at point C, determine (a) the speed at point B, (b) the magnitude of the total acceleration when t = 20 s.

Race car A is traveling on a straight portion of the track while race car B is traveling on a circular portion of the track. At the instant shown, the speed of A is increasing at the rate of \(10 \ \mathrm{m} / \mathrm{s}^{2}\), and the speed of B is decreasing at the rate of \(6 \ \mathrm{m} / \mathrm{s}^{2}\). For the position shown, determine (a) the velocity of B relative to A, (b) the acceleration of B relative to A.

At a given instant in an airplane race, airplane A is flying horizontally in a straight line, and its speed is being increased at the rate of \(8 \ \mathrm{m} / \mathrm{s}^{2}\). Airplane B is flying at the same altitude as airplane A and, as it rounds a pylon, is following a circular path of 300-m radius. Knowing that at the given instant the speed of B is being decreased at the rate of \(3 \mathrm{~m} / \mathrm{s}^{2}\), determine, for the positions shown, (a) the velocity of B relative to A, (b) the acceleration of B relative to A.

From a photograph of a homeowner using a snowblower, it is determined that the radius of curvature of the trajectory of the snow was 30 ft as the snow left the discharge chute at A. Determine (a) the discharge velocity \(v_{A}\) of the snow, (b) the radius of curvature of the trajectory at its maximum height.

A basketball is bounced on the ground at point A and rebounds with a velocity \(v_{A}\) of magnitude 7.5 ft/s as shown. Determine the radius of curvature of the trajectory described by the ball (a) at point A, (b) at the highest point of the trajectory.

A golfer hits a golf ball from point A with an initial velocity of 50 m/s at an angle of \(25^{\circ}\) with the horizontal. Determine the radius of curvature of the trajectory described by the ball (a) at point A, (b) at the highest point of the trajectory.

Three children are throwing snowballs at each other. Child A throws a snowball with a horizontal velocity \(v_{0}\). If the snowball just passes over the head of child B and hits child C, determine the radius of curvature of the trajectory described by the snowball (a) at point B, (b) at point C.

Coal is discharged from the tailgate A of a dump truck with an initial velocity \(\mathbf{v}_{A}=2 \ \mathrm{m} / \mathrm{s} \ \mathrm{d} \ 50^{\circ}\). Determine the radius of curvature of the trajectory described by the coal (a) at point A, (b) at the point of the trajectory 1 m below point A.

From measurements of a photograph, it has been found that as the stream of water shown left the nozzle at A, it had a radius of curvature of 25 m. Determine (a) the initial velocity \(v_{A}\) of the stream, (b) the radius of curvature of the stream as it reaches its maximum height at B.

A child throws a ball from point A with an initial velocity \(v_{A}\) of 20 m/s at an angle of \(25^{\circ}\) with the horizontal. Determine the velocity of the ball at the points of the trajectory described by the ball where the radius of curvature is equal to three-quarters of its value at A.

A projectile is fired from point A with an initial velocity \(v_{0}\). a) Show that the radius of curvature of the trajectory of the projectile reaches its minimum value at the highest point B of the trajectory. (b) Denoting by u the angle formed by the trajectory and the horizontal at a given point C, show that the radius of curvature of the trajectory at C is \(r=r_{\min } / \cos ^{3} u\).

Determine the radius of curvature of the path described by the particle of Prob. 11.95 when t = 0.

Determine the radius of curvature of the path described by the particle of Prob. 11.96 when t = 0, A = 3, and B = 1.

A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the acceleration of the satellite is equal to \(g(R / r)^{2}\), where g is the acceleration of gravity at the surface of the planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is \(274 \ \mathrm{m} / \mathrm{s}^{2}\), determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular. Earth: \(\left(v_{\text {mean }}\right)_{\text {orbit }}=107 \ \mathrm{Mm} / \mathrm{h}\).

A satellite will travel indefinitely in a circular orbit around a planet if the normal component of the acceleration of the satellite is equal to \(g(R / r)^{2}\), where g is the acceleration of gravity at the surface of the planet, R is the radius of the planet, and r is the distance from the center of the planet to the satellite. Knowing that the diameter of the sun is 1.39 Gm and that the acceleration of gravity at its surface is \(274 \ \mathrm{m} / \mathrm{s}^{2}\), determine the radius of the orbit of the indicated planet around the sun assuming that the orbit is circular. Saturn: \(\left(v_{\text {mean }}\right)_{\text {orbit }}=34.7 \ \mathrm{Mm} / \mathrm{h}\).

Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a circular orbit 100 mi above the surface of the planet. (See information given in Probs. 11.153–11.154.) Venus: \(g=29.20 \ \mathrm{ft} / \mathrm{s}^{2}, \quad R=3761 \mathrm{mi}\).

Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a circular orbit 100 mi above the surface of the planet. (See information given in Probs. 11.153–11.154.) Mars: \(g=12.17 \ \mathrm{ft} / \mathrm{s}^{2}, \quad R=2102 \ \mathrm{mi}\).

Determine the speed of a satellite relative to the indicated planet if the satellite is to travel indefinitely in a circular orbit 100 mi above the surface of the planet. (See information given in Probs. 11.153–11.154.) Jupiter: \(g=75.35 \ \mathrm{ft} / \mathrm{s}^{2}, \quad R=44,432 \ \mathrm{mi}\).

A satellite is traveling in a circular orbit around Mars at an altitude of 300 km. After the altitude of the satellite is adjusted, it is found that the time of one orbit has increased by 10 percent. Knowing that the radius of Mars is 3382 km, determine the new altitude of the satellite. (See information given in Probs. 11.153–11.154).

Knowing that the radius of the earth is 6370 km, determine the Problems time of one orbit of the Hubble Space Telescope knowing that the telescope travels in a circular orbit 590 km above the surface of the earth. (See information given in Probs. 11.153–11.154.)

Satellites A and B are traveling in the same plane in circular orbits around the earth at altitudes of 120 and 200 mi, respectively. If at t = 0 the satellites are aligned as shown and knowing that the radius of the earth is R = 3960 mi, determine when the satellites will next be radially aligned. (See information given in Probs. 11.153–11.154.)

The oscillation of rod OA about O is defined by the relation \(\mathrm{u}=(3 / p)(\sin \mathrm{p} t)\), where u and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from O is \(r=6\left(1-e^{-2 t}\right)\) where r and t are expressed in inches and seconds, respectively. When t = 1 s, determine (a) the velocity of the collar, (b) the acceleration of the collar, (c) the acceleration of the collar relative to the rod.

The rotation of rod OA about O is defined by the relation \(\mathrm{u}=t^{3}-4 t\), where u and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from O is \(r=2.5 t^{3}-5 t^{2}\), where r and t are expressed in inches and seconds, respectively. When t = 1 s, determine (a) the velocity of the collar, (b) the acceleration of the collar, (c) the radius of curvature of the path of the collar.

The path of particle P is the ellipse defined by the relations \(r=2 /(2-\cos \mathrm{p} t)\) and u = pt, where r is expressed in meters, t is in seconds, and u is in radians. Determine the velocity and the acceleration of the particle when (a) t = 0, (b) t = 0.5 s.

The two-dimensional motion of a particle is defined by the relations r = 2a cos u and \(\mathrm{u}=b t^{2} / 2\), where a and b are constants. Determine (a) the magnitudes of the velocity and acceleration at any instant, (b) the radius of curvature of the path. What conclusion can you draw regarding the path of the particle?

As rod OA rotates, pin P moves along the parabola BCD. Knowing that the equation of this parabola is \(r=2 b /(1+\cos u)\) and that u = kt, determine the velocity and acceleration of P when (a) u = 0, (b) \(\mathrm{u}=90^{\circ}\).

The pin at B is free to slide along the circular slot DE and along the rotating rod OC. Assuming that the rod OC rotates at a constant rate \(\dot{u}\), (a) show that the acceleration of pin B is of constant magnitude, (b) determine the direction of the acceleration of pin B.

To study the performance of a race car, a high-speed camera is positioned at point A. The camera is mounted on a mechanism which permits it to record the motion of the car as the car travels on straightaway BC. Determine (a) the speed of the car in terms of b, u, and \(\dot{u}\), (b) the magnitude of the acceleration in terms of b, u, \(\dot{u}\), and \(\ddot{\mathrm{u}}\).

After taking off, a helicopter climbs in a straight line at a constant angle b. Its flight is tracked by radar from point A. Determine the speed of the helicopter in terms of d, b, u, and \(\dot{\mathrm{u}}\).

At the bottom of a loop in the vertical plane an airplane has a horizontal velocity of 315 mi/h and is speeding up at a rate of \(10 \ \mathrm{ft} / \mathrm{s}^{2}\). The radius of curvature of the loop is 1 mi. The plane is being tracked by radar at O. What are the recorded values of \(\dot{r}\), \(\ddot{r}\), \(\dot{u}\), and \(\ddot{u}\) for this instant?

Pin C is attached to rod BC and slides freely in the slot of rod OA which rotates at the constant rate v. At the instant when \(b = 60^{\circ}\), determine (a) \(\dot{r}\) and \(\dot{u}\), (b) \(\ddot{r}\) and \(\ddot{u}\). Express your answers in terms of d and v.

For the race car of Prob. 11.167, it was found that it took 0.5 s for the car to travel from the position \(u = 60^{\circ}\) to the position \(u = 35^{\circ}\). Knowing that b = 25 m, determine the average speed of the car during the 0.5-s interval.

For the helicopter of Prob. 11.168, it was found that when the helicopter was at B, the distance and the angle of elevation of the helicopter were r = 3000 ft and \(u = 20^{\circ}\), respectively. Four seconds later, the radar station sighted the helicopter at r = 3320 ft and \(u = 23.1^{\circ}\). Determine the average speed and the angle of climb b of the helicopter during the 4-s interval.

A particle moves along the spiral shown; determine the magnitude of the velocity of the particle in terms of b, u, and \(\dot{u}\).

A particle moves along the spiral shown; determine the magnitude of the velocity of the particle in terms of b, u, and \(\dot{u}\).

A particle moves along the spiral shown. Knowing that \(\dot{u}\) is constant and denoting this constant by v, determine the magnitude of the acceleration of the particle in terms of b, u, and v.

A particle moves along the spiral shown. Knowing that \(\dot{u}\) is constant and denoting this constant by v, determine the magnitude of the acceleration of the particle in terms of b, u, and v.

The motion of a particle on the surface of a right circular cylinder is defined by the relations R = A, u = 2pt, and z = B sin 2pnt, where A and B are constants and n is an integer. Determine the magnitudes of the velocity and acceleration of the particle at any time t.

Show that \(\dot{r}=h \dot{\mathrm{f}} \ \sin \mathrm{u}\) knowing that at the instant shown, step AB of the step exerciser is rotating counterclockwise at a constant rate \(\dot{\mathrm{f}}\).

The three-dimensional motion of a particle is defined by the relations \(R=A\left(1-e^{-t}\right), \ \mathrm{u}=2 \mathrm{p} t, \text { and } z=B\left(1-e^{-t}\right)\). Determine the magnitudes of the velocity and acceleration when (a) t = 0, (b) \(t=\infty\).

For the conic helix of Prob. 11.95, determine the angle that the osculating plane forms with the y axis.

Determine the direction of the binormal of the path described by the particle of Prob. 11.96 when (a) t = 0, (b) t = p/2 s.

The motion of a particle is defined by the relation \(x=2 t^{3}-15 t^{2}+24 t+4\), where x and t are expressed in meters and seconds, respectively. Determine (a) when the velocity is zero, (b) the position and the total distance traveled when the acceleration is zero.

A particle starting from rest at x = 1 m is accelerated so that its velocity doubles in magnitude between x = 2 m and x = 8 m. Knowing that the acceleration of the particle is defined by the relation a = k[x - (A/x)], determine the values of the constants A and k if the particle has a velocity of 29 m/s when x = 16 m.

A particle moves in a straight line with the acceleration shown in the figure. Knowing that the particle starts from the origin with \(v_{0}=-2 \ \mathrm{m} / \mathrm{s}\), (a) construct the v–t and x–t curves for 0 < t < 18 s, (b) determine the position and the velocity of the particle and the total distance traveled when t = 18 s.

The velocities of commuter trains A and B are as shown. Knowing that the speed of each train is constant and that B reaches the crossing 10 min after A passed through the same crossing, determine (a) the relative velocity of B with respect to A, (b) the distance between the fronts of the engines 3 min after A passed through the crossing.

Slider block B starts from rest and moves to the right with a constant acceleration of \(1 \ \mathrm{ft} / \mathrm{s}^{2}\). Determine (a) the relative acceleration of portion C of the cable with respect to slider block A, (b) the velocity of portion C of the cable after 2 s.

Collar A starts from rest at t = 0 and moves downward with a constant acceleration of \(7 \text { in./s }{ }^{2}\). Collar B moves upward with a constant acceleration, and its initial velocity is 8 in./s. Knowing that collar B moves through 20 in. between t = 0 and t = 2 s, determine (a) the accelerations of collar B and block C, (b) the time at which the velocity of block C is zero, (c) the distance through which block C will have moved at that time.

A golfer hits a ball with an initial velocity of magnitude \(v_{0}\) at an angle a with the horizontal. Knowing that the ball must clear the tops of two trees and land as close as possible to the flag, determine \(v_{0}\) and the distance d when the golfer uses (a) a six-iron with \(a=31^{\circ}\), (b) a five-iron with \(\mathrm{a}=27^{\circ}\).

As the truck shown begins to back up with a constant acceleration of \(4 \ \mathrm{ft} / \mathrm{s}^{2}\), the outer section B of its boom starts to retract with a constant acceleration of \(1.6 \ \mathrm{ft} / \mathrm{s}^{2}\) relative to the truck. Determine (a) the acceleration of section B, (b) the velocity of section B when t = 2 s.

A motorist traveling along a straight portion of a highway is decreasing the speed of his automobile at a constant rate before exiting from the highway onto a circular exit ramp with a radius of 560 ft. He continues to decelerate at the same constant rate so that 10 s after entering the ramp, his speed has decreased to 20 mi/h, a speed which he then maintains. Knowing that at this constant speed the total acceleration of the automobile is equal to one-quarter of its value prior to entering the ramp, determine the maximum value of the total acceleration of the automobile.

Sand is discharged at A from a conveyor belt and falls onto the top of a stockpile at B. Knowing that the conveyor belt forms an angle \(\mathrm{a}=25^{\circ}\) with the horizontal, determine (a) the speed \(v_{0}\) of the belt, (b) the radius of curvature of the trajectory described by the sand at point B.

The end point B of a boom is originally 5 m from fixed point A when the driver starts to retract the boom with a constant radial acceleration of \(\ddot{r}=-1.0 \ \mathrm{m} / \mathrm{s}^{2}\) and lower it with a constant angular acceleration \(\ddot{\mathrm{u}}=-0.5 \ \mathrm{rad} / \mathrm{s}^{2}\). At t = 2 s, determine (a) the velocity of point B, (b) the acceleration of point B, (c) the radius of curvature of the path.

A telemetry system is used to quantify kinematic values of a ski jumper immediately before she leaves the ramp. According to the system \(r=500 \ \mathrm{ft}, \ \dot{r}=-105 \ \mathrm{ft} / \mathrm{s}, \ \ddot{r}=-10 \mathrm{ft} / \mathrm{s}^{2}, \ \mathrm{u}=25^{\circ}, \ \dot{\mathrm{u}}=0.07 \ \mathrm{rad} / \mathrm{s}, \ \ddot{\mathrm{u}}=0.06 \ \mathrm{rad} / \mathrm{s}^{2}\). Determine (a) the velocity of the skier immediately before she leaves the jump, (b) the acceleration of the skier at this instant, (c) the distance of the jump d neglecting lift and air resistance.