A cannon on a train car fires a projectile to the right

An amusement park game, shown in Figure CP4.86, launches a marble toward a small cup. The marble is placed directly on top of a spring-loaded wheel and held with a clamp. When released, the wheel spins around clockwise at constant angular acceleration, opening the clamp and releasing the marble after making \(\frac{11}{12}\) revolution. What angular acceleration is needed for the ball to land in the cup? The top of the cup is level with the center of the wheel. Equation Transcription: Text Transcription: 11/12

Problem 85CP A child in danger of drowning in a river is being carried downstream by a current that flows uniformly with a speed of 2.0 m/s. The child is 200 m from the shore and 1500 m upstream of the boat dock from which the rescue team sets out. If their boat speed is 8.0 m/s with respect to the water, at what angle from the shore should the pilot leave the shore to go directly to the child?

1. At this instant, is the particle in Figure Q4.1 speeding up, slowing down, or traveling at constant speed? 2. Is this particle curving to the right, curving to the left, or traveling straight?

1. At this instant, is the particle in Figure Q4.2 speeding up, slowing down, or traveling at constant speed? 2. Is this particle curving upward, curving downward, or traveling straight?

Problem 84CP A cannon on a train car fires a projectile to the right with speed v0, relative to the train, from a barrel elevated at angle ?. The cannon fires just as the train, which had been causing to the right along a level track with speed vtrain, begins to accelerate with acceleration a. Find an expression for the angle at which the projectile should be fired so that it lands as far as possible from the cannon. You can ignore the small height of the cannon above the track.

Problems 1 and 2 show a partial motion diagram. For each: 1. Complete the motion diagram by adding acceleration vectors. 2. Write a physics problem for which this is the correct motion diagram. Be imaginative! Don’t forget to include enough information to make the problem complete and to state clearly what is to be found.

Problems 1 and 2 show a partial motion diagram. For each: 1. Complete the motion diagram by adding acceleration vectors. 2. Write a physics problem for which this is the correct motion diagram. Be imaginative! Don’t forget to include enough information to make the problem complete and to state clearly what is to be found.

Tarzan swings through the jungle by hanging from a vine. 1. Immediately after stepping off a branch to swing over to another tree, is Tarzan’s acceleration au zero or not zero? If not zero, which way does it point? Explain. 2. Answer the same question at the lowest point in Tarzan’s swing.

For a projectile, which of the following quantities are constant during the flight: \(x, y, r, vx , vy , v, ax , ay\)? Which of these quantities are zero throughout the flight? Equation Transcription: Text Transcription: x, y, r, vx , vy , v, ax , ay

A projectile is launched at an angle of \(30^{\circ}\). a. Is there any point on the trajectory where \(\vec{v}\) and \(\vec{a}\) are parallel to each other? If so, where? b. Is there any point where \(\vec{v}\) and \(\vec{a}\) are perpendicular to each other? If so, where? Equation Transcription: Text Transcription: 30^circ ^vec v ^vec a ^vec v ^vec a

At this instant, the particle is slowing and curving upward. What is the direction of its acceleration?

At this instant, the particle is speeding up and curving downward. What is the direction of its acceleration?

At this instant, the particle has steady speed and is curving to the right. What is the direction of its acceleration?

Problem 6CQ A cart that is rolling at constant velocity on a level table fires a ball straight up. a. When the ball comes back down, will it land in front of the launching tube, behind the launching tube, or directly in the tube? Explain. ________________ b. Will your answer change if the cart is accelerating in the forward direction? If so, how?

A sailboat is traveling east at 5.0 m/s. A sudden gust of wind gives the boat an acceleration = (0.80 m/s2, 40° north of east). What are the boat’s speed and direction 6.0 s later when the gust subsides?

Problem 7E Section 4.2 Two-Dimensional Kinematics A model rocket is launched from rest with an upward acceleration of 6.00 m/s2 and, due to a strong wind, a horizontal acceleration of 1.50 m/s2 How far is the rocket from the launch pad 6.00 s later when the rocket engine runs out of fuel?

Problem 7CQ A rock is thrown from a bridge at an angle 30° below horizontal. Immediately after the rock is released, is the magnitude of its acceleration greater than, less than, or equal to g? Explain.

Anita is running to the right at \(5 \mathrm{~m} / \mathrm{s}\) in Figure Q4.8. Balls 1 and 2 are thrown toward her by friends standing on the ground. According to Anita, both balls are approaching her at \(10 \mathrm{~m} / \mathrm{s}\). Which ball was thrown at a faster speed? Or were they thrown with the same speed? Explain. Equation Transcription: Text Transcription: 5m/s 10 m/s

Problem 9CQ An electromagnet on the ceiling of an airplane holds a steel ball. When a button is pushed, the magnet releases the ball. The experiment is first done while the plane is parked on the ground, and the point where the ball hits the floor is marked with an X. Then the experiment is repeated while the plane is flying level at a steady 500 mph. Does the ball land slightly in front of the X (toward the nose of the plane), on the X, or slightly behind the X (toward the tail of the plane)? Explain.

A rocket-powered hockey puck moves on a horizontal frictionless table. Figure EX 4.9 shows graphs of \(v_{x}\) and \(v_{y}\), the \(x\) - and \(y\)-components of the puck's velocity. The puck starts at the origin. a. In which direction is the puck moving at \(t=2\) s? Give you answer as an angle from the \(x\)-axis. b. How far from the origin is the puck at \(t=5 \mathrm{~s}\)? Equation Transcription: Text Transcription: v_x v_y x y-components t=2 s x-axis t=5 s

Zack is driving past his house in Figure Q4.10. He wants to toss his physics book out the window and have it land in his driveway. If he lets go of the book exactly as he passes the end of the driveway, should he direct his throw outward and toward the front of the car (throw 1), straight outward (throw 2), or outward and toward the back of the car (throw 3)? Explain.

A particle's trajectory is described by \(x=\left(\frac{1}{2} t^{3}-2 t^{2}\right) \mathrm{m}\) and \(y=\left(\frac{1}{2} t^{2}-2 t\right) \mathrm{m}\), where \(t\) is in s. a. What are the particle's position and speed at \(t=0 \mathrm{~s}\) and \(t=4 \mathrm{~s}\)? b. What is the particle's direction of motion, measured as an angle from the \(x\)-axis, at \(t=0 \mathrm{~s}\) and \(t=4 \mathrm{~s}\)? Equation Transcription: )m Text Transcription: x=(1/2 t^3-2 t^2)m y=(1/2 t^2-2t)m t t=0 s t= 4 s x-axis t=0 s t= 4 s

A rocket-powered hockey puck moves on a horizontal frictionless table. FIGURE EX4.10 shows graphs of \(v_{x}\) and \(v_{y}\), the \(x\)-and \(y\)-components of the puck's velocity. The puck starts at the origin. What is the magnitude of the puck's acceleration at \(t=5 \mathrm{~s}\) ? Equation Transcription: Text Transcription: v_x V_ y x y-components t=5 s

In Figure Q4.11, Yvette and Zack are driving down the freeway side by side with their windows down. Zack wants to toss his physics book out the window and have it land in Yvette’s front seat. Ignoring air resistance, should he direct his throw outward and toward the front of the car (throw 1), straight outward (throw 2), or outward and toward the back of the car (throw 3)? Explain.

A physics student on Planet Exidor throws a ball, and it follows the parabolic trajectory shown in FIGURE EX4.11. The ball's position is shown at \(1 \mathrm{~s}\) intervals until \(t=3 \mathrm{~s}\). At \(t=1 \mathrm{~s}\), the ball's velocity is \(\vec{v}=(2.0 \hat{\imath}+2.0 \hat{\jmath}) \mathrm{m} / \mathrm{s}\). a. Determine the ball's velocity at \(t=0 \mathrm{~s}, 2 \mathrm{~s}\), and \(3 \mathrm{~s}\). b. What is the value of \(g\) on Planet Exidor? c. What was the ball's launch angle? Equation Transcription: Text Transcription: 1 s t=3 s t=1 s ^vec v=(2.0 ^hat i+2.0 ^hat j) m /s t=0 s, 2 s, 3 s g

Problem 12CQ In uniform circular motion, which of the following quantities are constant: speed, instantaneous velocity, tangential velocity, radial acceleration, tangential acceleration? Which of these quantities are zero throughout the motion?

Problem 12E Section 4.3 Projectile Motion A ball thrown horizontally at 25 m/s travels a horizontal distance of 50 m before hitting the ground. From what height was the ball thrown?

Problem 13E A rifle is aimed horizontally at a target 50 m away. The bullet hits the target 2.0 cm below the aim point. a. What was the bullet’s flight time? b. What was the bullet’s speed as it left the barrel?

Problem 15E Section 4.4 Relative Motion A boat takes 3.0 hours to travel 30 km down a river, then 5.0 hours to return. How fast is the river flowing?

FIGURE Q4.13 shows three points on a steadily rotating wheel. a. Rank in order, from largest to smallest, the angular velocities \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) of these points. Explain. b. Rank in order, from largest to smallest, the speeds \(v_{1}, v_{2}\), and \(v_{3}\) of these points. Explain. Equation Transcription: Text Transcription: omega_1,omega_2,omega_3 v_1,v_2,v_3

FIGURE Q4.15 shows a pendulum at one end point of its arc. a. At this point, is \(\omega\) positive, negative, or zero? Explain. b. At this point, is \(\alpha\) positive, negative, or zero? Explain. Equation Transcription: Text Transcription: Omega alpha

FIGURE Q4.14 shows four rotating wheels. For each, determine the signs \((+ or -)\) of \(\omega\) and \(\alpha\). Equation Transcription: Text Transcription: (+ or -) Omega alpha

Problem 14E Section 4.3 Projectile Motion A supply plane needs to drop a package of food to scientists working on a glacier in Greenland. The plane flies 100 m above the glacier at a speed of 150 m/s. How far short of the target should it drop the package?

Problem 16E Section 4.4 Relative Motion When the moving sidewalk at the airport is broken, as it often seems to be, it takes you 50 s to walk from your gate to baggage claim. When it is working and you stand on the moving sidewalk the entire way, without walking, it takes 75 s to travel the same distance. How long will it take you to travel from the gate to baggage claim if you walk while riding on the moving sidewalk?

Problem 17E Section 4.4 Relative Motion Mary needs to row her boat across a 100-m-wide river that is flowing to the east at a speed of 1.0 m/s. Mary can row with a speed of 2.0 m/s. a. If Mary points her boat due north, how far from her intended landing spot will she be when she reaches the opposite shore? ________________ b. What is her speed with respect to the shore?

FIGURE EX4.20 shows the angular-velocity-versus-time graph for a particle moving in a circle. How many revolutions does the object make during the first \(4 s\)? Equation Transcription: Text Transcription: 4 s

FIGURE EX4.19 shows the angular-position-versus-time graph for a particle moving in a circle. What is the particle's angular velocity at (a) \(t=1 \mathrm{~s}\), (b) \(t=4 \mathrm{~s}\), and (c) \(t=7 \mathrm{~s}\)? Equation Transcription: Text Transcription: t=1 s t=4 s t=7 s

Problem 18E Section 4.4 Relative Motion Susan, driving north at 60 mph, and Trent, driving east at 45 mph, are approaching an intersection. What is Trent’s speed relative to Susan’s reference frame?

Problem 23E Section 4.5 Uniform Circular Motion The earth’s radius is about 4000 miles. Kampala, the capital of Uganda, and Singapore are both nearly on the equator. The distance between them is 5000 miles. The flight from Kampala to Singapore takes 9.0 hours. What is the plane’s angular velocity with respect to the earth’s surface? Give your answer in °/h.

Problem 22E Section 4.5 Uniform Circular Motion An old-fashioned single-play vinyl record rotates on a turntable at 45 rpm. What are (a) the angular velocity in rad/s and (b) the period of the motion?

FIGURE EX4.21 shows the angular-velocity-versus-time graph for a particle moving in a circle, starting from \(\theta_{0}=0 \mathrm{rad}\) at \(t=0 \mathrm{~s}\). Draw the angular-position-versus-time graph. Include an appropriate scale on both axes. Equation Transcription: Text Transcription: theta_0=0 t=0 s

A particle starts from rest at \(\vec{r}_{0}=9.0 \hat{\jmath} \mathrm{m}\) and moves in the \(x y\)-plane with the velocity shown in FIGURE P4.36. The particle passes through a wire hoop located at \(\vec{r}_{1}=201^{\hat{n}} m\), then continues onward. a. At what time does the particle pass through the hoop? b. What is the value of \(v_{4 y^{\prime}}\) the \(y\)-component of the particle's velocity at \(t=4 \mathrm{~s}\) Equation Transcription: Text Transcription: ^vec r_0=9.0 ^hat j xy-plane ^vec r_1=20 ^hat i m v_4y y-component t=4 s

A spaceship maneuvering near Planet Zeta is located at \(\vec{r}=(600 \hat{\imath}-400 \hat{\jmath}+200 \hat{k}) \times 10^{3} \mathrm{~km}\), relative to the planet, and traveling at \(\vec{v}=9500 \hat{\imath} \mathrm{m} / \mathrm{s}\). It turns on its thruster engine and accelerates with \(\vec{a}=(40 \hat{\imath}-20 \hat{k}) \mathrm{m} / \mathrm{s}^{2}\) for \(35 \mathrm{~min}\). Where is the spaceship located when the engine shuts off? Give your answer as a vector measured in km. Equation Transcription: Text Transcription: ^vec r=(600 ^hat i-400 ^hat j+200 ^hat k)10^3km ^vec v=9500 ^hat im/s ^vec a=(40 ^hat i -20 ^hat k)m/s^2 35 min .

A projectile's horizontal range on level ground is \(\mathrm{R}=v_{0}^{2} \ \sin 2 \theta / \mathrm{g}\). At what launch angle or angles will the projectile land at half of its maximum possible range? Equation Transcription: R = sin 2/g Text Transcription: R = v _0 ^2 sin 2 theta/g

Problem 69P A Ferris wheel of radius R speeds up with angular acceleration ? starting from rest. Find an expression for the (a) velocity and (b) centripetal acceleration of a rider after the Ferris wheel has rotated through angle ??.

A \(6.0\)-cm-diameter gear rotates with angular velocity \(\omega=\left(2.0+\frac{1}{2} t^{2}\right) \mathrm{rad} / \mathrm{s}\), where \(t\) is in seconds. At \(t=4.0 \mathrm{~s}\), what are: a. The gear's angular acceleration? b. The tangential acceleration of a tooth on the gear? Equation Transcription: Text Transcription: 6.0 cm omega (2.0+1/2 t^2)rad/s t t=4.0 s

Problem 71P On a lonely highway, with no other cars in sight, you decide to measure the angular acceleration of your engine’s crankshaft while braking gently. Having excellent memory, you are able to read the tachometer every 1.0 s and remember seven values long enough to later write them down. The table shows your data: Time (s) rpm 0.0 3010 1.0 2810 2.0 2450 3.0 2250 4.0 1940 5.0 1810 6.0 1510 What is the magnitude of the crankshaft’s angular acceleration? Give your result in rad/s2.

a. A projectile is launched with speed \(v_{0}\) and angle \(\theta\). Derive an expression for the projectile’s maximum height h. b. A baseball is hit with a speed of 33.6 m/s. Calculate its height and the distance traveled if it is hit at angles of 30.0°, 45.0°, and 60.0°.

A gray kangaroo can bound across level ground with each jump carrying it \(10 \mathrm{~m}\) from the takeoff point. Typically the kangaroo leaves the ground at a \(20^{\circ}\) angle. If this is so: a. What is its takeoff speed? b. What is its maximum height above the ground? Equation Transcription: Text Transcription: 10 m 20^circ

Problem 41P A projectile is fired with an initial speed of 30 m/s at an angle of 60° above the horizontal. The object hits the ground 7.5 s later. a. How much higher or lower is the launch point relative to the point where the projectile hits the ground? ________________ b. To what maximum height above the launch point does the projectile rise?

Problem 72P A car starts from rest on a curve with a radius of 120 m and accelerates at 1.0 m/s2. Through what angle will the car have traveled when the magnitude of its total acceleration is 2.0 m/s2?

Problem 73P A long string is wrapped around a 6.0-cm-diameter cylinder, initially at rest, that is free to rotate on an axle. The string is then pulled with a constant acceleration of 1.5 m/s2 until 1.0 m of string has been unwound. If the string unwinds without slipping, what is the cylinder’s angular speed, in rpm, at this time?

In Problems 74 through 76 you are given the equations that are used to solve a problem. For each of these, you are to a. Write a realistic problem for which these are the correct equations. Be sure that the answer your problem requests is consistent with the equations given. b. Finish the solution of the problem, including a pictorial representation. \(100 \ \mathrm{m}=0 \ \mathrm{m}+(50 \ \mathrm{cos} \ \theta \ \mathrm{m} / \mathrm{s}) \mathrm{t}_{1}\) \(0 \ \mathrm{m}=0 \ \mathrm{m}+(50 \sin \theta \ \mathrm{m} / \mathrm{s}) \mathrm{t}_{1}-\frac{1}{2}\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right) \mathrm{t}_{1} \ ^{2}\) Equation Transcription: 100 m = 0 m + (50 cos m/s)t1 0 m = 0 m + (50 sin m/s)t1 - (9.80 m/s2)t1 2 Text Transcription: 100 m = 0 m + (50 cos theta m/s)t_1 0 m = 0 m + (50 sin theta m/s)t_1 - 1/2 (9.80 m/s^2)t_1 ^2

Problem 42P In the Olympic shotput event, an athlete throws the shot with an initial speed of 12.0 m/s at a 40.0° angle from the horizontal. The shot leaves her hand at a height of 1.80 m above the ground. a. How far does the shot travel? ________________ b. Repeat the calculation of part (a) for angles 42.5°, 45.0°, and 47.5°. Put all your results, including 40.0°, in a table. At what angle of release does she throw the farthest?

Problem 43P On the Apollo 14 mission to the moon, astronaut Alan Shepard hit a golf ball with a golf club improvised from a tool. The free-fall acceleration on the moon is 1/6 of its value on earth. Suppose he hit the ball with a speed of 25 m/s at an angle 30° above the horizontal. a. How long was the ball in flight? b. How far did it travel? c. Ignoring air resistance, how much farther would it travel on the moon than on earth?

Problem 44P A ball is thrown toward a cliff of height h with a speed of 30 m/s and an angle of 60° above horizontal. It lands on the edge of the cliff 4.0 s later. a. How high is the cliff? ________________ b. What was the maximum height of the ball? ________________ c. What is the ball’s impact speed?

Problem 75P In Problem you are given the equations that are used to solve a problem. For these, you are to a. Write a realistic problem for which these are the correct equations. Be sure that the answer your problem requests is consistent with the equations given. ________________ b. Finish the solution of the problem, including a pictorial representation. vx = ?(6.0cos45°) m/s + 3.0 m/s vy = (6.0 sin 45°) m/s + 0 m/s 100 m = vyt1, x1 = vxt1

Problem 76P In Problem you are given the equations that are used to solve a problem. For these, you are to a. Write a realistic problem for which these are the correct equations. Be sure that the answer your problem requests is consistent with the equations given. ________________ b. Finish the solution of the problem, including a pictorial representation. 2.5 rad = 0 rad + ?i(10s)+ ((1.5 m/s2)/2(50 m))(10 s)2 ?f = ?i + ((1.5 m/s2)/(50 m) ) (10 s)

You are asked to consult for the city's research hospital, where a group of doctors is investigating the bombardment of cancer tumors with high-energy ions. The ions are fired directly toward the center of the tumor at speeds of \(5.0 \times 10^{6} \mathrm{~m} / \mathrm{s}\). To cover the entire tumor area, the ions are deflected sideways by passing them between two charged metal plates that accelerate the ions perpendicular to the direction of their initial motion. The acceleration region is \(5.0 \mathrm{~cm}\) long, and the ends of the acceleration plates are \(1.5 \mathrm{~m}\) from the patient. What sideways acceleration is required to deflect an ion \(2.0 \mathrm{~cm}\) to one side? Equation Transcription: Text Transcription: 5.0 x 10^6m/s 5.0 cm 1.5 m 2.0 cm

Problem 45P A tennis player hits a ball 2.0 m above the ground. The ball leaves his racquet with a speed of 20 m/s at an angle 5.0° above the horizontal. The horizontal distance to the net is 7.0 m, and the net is 1.0 m high. Does the ball clear the net? If so, by how much? If not, by how much does it miss?

Problem 46P A baseball player friend of yours wants to determine his pitching speed. You have him stand on a ledge and throw the ball horizontally from an elevation 4.0 m above the ground. The ball lands 25 m away. a. What is his pitching speed? ________________ b. As you think about it, you’re not sure he threw the ball exactly horizontally. As you watch him throw, the pitches seem to vary from 5° below horizontal to 5° above horizontal. What are the lowest and highest speeds with which the ball might have left his hand?

Problem 47P You are playing right field for the baseball team. Your team is up by one run in the bottom of the last inning of the game when a ground ball slips through the infield and comes straight toward you. As you pick up the ball 65 m from home plate, you see a runner rounding third base and heading for home with the tying run. You throw the ball at an angle of 30° above the horizontal with just the right speed so that the ball is caught by the catcher, standing on home plate, at the same height as you threw it. As you release the ball, the runner is 20 m from home plate and running full speed at 8.0 m/s. Will the ball arrive in time for your team’s catcher to make the tag and win the game?

Problem 79CP You are watching an archery tournament when you start wondering how fast an arrow is shot from the bow. Remembering your physics, you ask one of the archers to shoot an arrow parallel to the ground. You find the arrow stuck in the ground 60 m away, making a 3.0° angle with the ground. How fast was the arrow shot?

In one contest at the county fair, seen in FIGURE CP4.78, a spring loaded plunger launches a ball at a speed of \(3.0 \mathrm{~m} / \mathrm{s}\) from one corner of a smooth, flat board that is tilted up at a \(20^{\circ}\) angle. To win, you must make the ball hit a small target at the adjacent corner, \(2.50 \mathrm{~m}\) away. At what angle \(\theta\) should you tilt the ball launcher? Equation Transcription: Text Transcription: 3.0 m/s 20^circ 2.50 m theta

An archer standing on a 15 slope shoots an arrow 20 above the horizontal, as shown in Figure CP4.80. How far down the slope does the arrow hit if it is shot with a speed of \(50 \mathrm{~m} / \mathrm{s} \mathrm{from} 1.75\) \(\mathrm{m}\) above the ground? Equation Transcription: Text Transcription: 50 m/s 1.75 m

You're \(6.0 \mathrm{~m}\) from one wall of the house seen in figure P4.48. You want to toss a ball to your friend who is \(6.0 \mathrm{~m}\) from the opposite wall. The throw and catch each occur \(1.0 \mathrm{~m}\) above the ground. a. What minimum speed will allow the ball to clear the roof? b. At what angle should you toss the ball? Equation Transcription: Text Transcription: 6.0 m 6.0 m 1.0 m

Sand moves without slipping at \(6.0 \mathrm{~m} / \mathrm{s}\) down a conveyer that is tilted at \(15^{\circ} .\) The sand enters a pipe \(3.0 \mathrm{~m}\) below the end of the conveyor belt, as shown in FIGURE P4.49. What is the horizontal distance \(d\) between the conveyor belt and the pipe? Equation Transcription: Text Transcription: 6.0 m/s 15^circ 3.0 m d

Problem 50P A stunt man drives a car at a speed of 20 m/s off a 30-m-high cliff. The road leading to the cliff is inclined upward at an angle of 20°. a. How far from the base of the cliff does the car land? ________________ b. What is the car’s impact speed?

A rubber ball is dropped onto a ramp that is tilted at 20 , as shown in FIGURE CP4.81. A bouncing ball obeys the "law of reflection," which says that the ball leaves the surface at the same angle it approached the surface. The ball's next bounce is \(3.0 \mathrm{~m}\) to the right of its first bounce. What is the ball's rebound speed on its first bounce? Equation Transcription: Text Transcription: 3.0 m

Problem 82CP A skateboarder starts up a 1.0-m-high, 30° ramp at a speed of 7.0 m/s. The skateboard wheels roll without friction. How far from the end of the ramp does the skateboarder touch down?

Problem 83CP A motorcycle daredevil wants to set a record for jumping over burning school buses. He has hired you to help with the design. He intends to ride off a horizontal platform at 40 m/s, cross the burning buses in a pit below him, then land on a ramp sloping down at 20°. It’s very important that he not bounce when he hits the landing ramp because that could cause him to lose control and crash. You immediately recognize that he won’t bounce if his velocity is parallel to the ramp as he touches down. This can be accomplished if the ramp is tangent to his trajectory and if he lands right on the front edge of the ramp. There’s no room for error! Your task is to determine where to place the landing ramp. That is, how far from the edge of the launching platform should the front edge of the landing ramp be horizontally and how far below it? There’s a clause in your contract that requires you to test your design before the hero goes on national television to set the record.

A javelin thrower standing at rest holds the center of the javelin behind her head, then accelerates it through a distance of \(70 \mathrm{~cm}\) as she throws. She releases the javelin \(2.0 \mathrm{~m}\) above the ground traveling at an angle of \(30^{\circ}\) above the horizontal. Toprated javelin throwers do throw at about a \(30^{\circ}\) angle, not the 45 you might have expected, because the biomechanics of the arm allow them to throw the javelin much faster at \(30^{\circ}\) than they would be able to at \(45^{\circ}\). In this throw, the javelin hits the ground \(62 \mathrm{~m}\) away. What was the acceleration of the javelin during the throw? Assume that it has a constant acceleration. Equation Transcription: Text Transcription: 70 cm 2.0 m 30^circ 30^circ 45^circ 62 m

Problem 52P Ships A and B leave port together. For the next two hours, ship A travels at 20 mph in a direction 30° west of north while ship B travels 20° east of north at 25 mph. a. What is the distance between the two ships two hours after they depart? b. What is the speed of ship A as seen by ship B?

A kayaker needs to paddle north across a 100-m-wide harbor. The tide is going out, creating a tidal current flowing east at 2.0 m/s. The kayaker can paddle with a speed of 3.0 m/s. a. In which direction should he paddle in order to travel straight across the harbor? b. How long will it take him to cross?

Problem 54P Mike throws a ball upward and toward the east at a 63° angle with a speed of 22 m/s. Nancy drives east past Mike at 30 m/s at the instant he releases the ball. a. What is the ball's initial angle in Nancy's reference frame? ________________ b. Find and graph the ball’s trajectory as seen by Nancy.

Problem 55P While driving north at 25 m/s during a rainstorm you notice that the rain makes an angle of 38° with the vertical. While driving back home moments later at the same speed but in the opposite direction, you see that the rain is falling straight down. From these observations, determine the speed and angle of the raindrops relative to the ground.

Problem 56P You’ve been assigned the task of using a shaft encoder—a device that measures the angle of a shaft or axle and provides a signal to a computer—to analyze the rotation of an engine crankshaft under certain conditions. The table lists the crankshaft's angles over a 0.6 s interval. Time (s) Angle (rad) 0.0 0.0 0.1 2.0 0.2 3.2 0.3 4.3 0.4 5.3 0.5 6.1 0.6 7.0 Is the crankshaft rotating with uniform circular motion? If so, what is its angular velocity in rpm? If not, is the angular acceleration positive or negative?

Problem 24E Section 4.6 Velocity and Acceleration in Uniform Circular Motion A 3000-m-high mountain is located on the equator. How much faster does a climber on top of the mountain move than a surfer at a nearby beach? The earth’s radius is 6400 km.

Problem 25E Section 4.6 Velocity and Acceleration in Uniform Circular Motion How fast must a plane fly along the earth’s equator so that the sun stands still relative to the passengers? In which direction must the plane fly, east to west or west to east? Give your answer in both km/h and mph. The earth’s radius is 6400 km.

Problem 26E Section 4.6 Velocity and Acceleration in Uniform Circular Motion To withstand “g-forces” of up to 10 g’s, caused by suddenly pulling out of a steep dive, fighter jet pilots train on a “human centrifuge.” 10 g’s is an acceleration of 98 m/s2. If the length of the centrifuge arm is 12 m, at what speed is the rider moving when she experiences 10 g’s?

Problem 57P A speck of dust on a spinning DVD has a centripetal acceleration of 20 m/s2. a. What is the acceleration of a different speck of dust that is twice as far from the center of the disk? ________________ b. What would be the acceleration of the first speck of dust if the disk’s angular velocity was doubled?

Problem 58P A typical laboratory centrifuge rotates at 4000 rpm. Test tubes have to be placed into a centrifuge very carefully because of the very large accelerations. a. What is the acceleration at the end of a test tube that is 10 cm from the axis of rotation? ________________ b. For comparison, what is the magnitude of the acceleration a test tube would experience if dropped from a height of 1.0 m and stopped in a 1.0-ms-long encounter with a hard floor?

Problem 59P Astronauts use a centrifuge to simulate the acceleration of a rocket launch. The centrifuge takes 30 s to speed up from rest to its top speed of 1 rotation every 1.3 s. The astronaut is strapped into a seat 6.0 m from the axis. a. What is the astronaut’s tangential acceleration during the first 30 s? ________________ b. How many g’s of acceleration does the astronaut experience when the device is rotating at top speed? Each 9.8 m/s2 of acceleration is 1 g.

Problem 27E Section 4.6 Velocity and Acceleration in Uniform Circular Motion The radius of the earth’s very nearly circular- orbit around the sun is 1.5 × 1011 m. Find the magnitude of the earth’s (a) velocity, (b) angular velocity, and (c) centripetal acceleration as it travels around the sun. Assume a year of 365 days.

Problem 28E Section 4.6 Velocity and Acceleration in Uniform Circular Motion Your roommate is working on his bicycle and has the bike upside down. He spins the 60-cm-diameter wheel, and you notice that a pebble stuck in the tread goes by three times every second. What are the pebble’s speed and acceleration?

FIGURE EX4.29 shows the angular velocity graph of the crankshaft in a car. What is the crankshaft's angular acceleration at (a) \(t=1 \mathrm{~s}\), (b) \(t=3 \mathrm{~s}\), and (c) \(t=5 \mathrm{~s}\)? Equation Transcription: Text Transcription: t=1 s t=3 s t= 5 s

Peregrine falcons are known for their maneuvering ability. In a tight circular turn, a falcon can attain a centripetal acceleration 1.5 times the free-fall acceleration. What is the radius of the turn if the falcon is flying at \(25 \mathrm{~m} / \mathrm{s}\)? Equation Transcription: Text Transcription: 25 m/s

Problem 61P As the earth rotates, what is the speed of (a) a physics student in Miami, Florida, at latitude 26°, and (b) a physics student in Fairbanks, Alaska, at latitude 65°? Ignore the revolution of the earth around the sun. The radius of the earth is 6400 km.

Problem 62P Communications satellites are placed in a circular orbit where they stay directly over a fixed point on the equator as the earth rotates. These are called geosynchronous orbits. The radius of the earth is 6.37 × 106 m, and the altitude of a geosynchronous orbit is 3.58 × 107 m (? 22,000 miles). What are (a) the speed and (b) the magnitude of the acceleration of a satellite in a geosynchronous orbit?

FIGURE EX4.31 shows the angular-velocity-versus-time graph for a particle moving in a circle. How many revolutions does the object make during the first \(4 \mathrm{~s}\)? Equation Transcription: Text Transcription: 4 s

Figure EX 4.30 shows the angular acceleration graph of a turntable that starts from rest. What is the turntable's angular velocity at (a) \(t=1 \mathrm{~s}\), (b) \(t=2 \mathrm{~s}\), and (c) \(t=3 \mathrm{~s}\) ? Equation Transcription: Text Transcription: t=1 s t=2 s t=3 s

Problem 32E Section 4.7 Nonuniform Circular Motion and Angular Acceleration A 5.0-m-diameter merry-go-round is initially turning with a 4.0 s period. It slows down and stops in 20 s. a. Before slowing, what is the speed of a child on the rim? ________________ b. How many revolutions does the merry-go-round make as it stops?

A computer hard disk \(8.0 \mathrm{~cm}\) in diameter is initially at rest. A small dot is painted on the edge of the disk. The disk accelerates at \(600 \mathrm{rad} / \mathrm{s}^{2}\) for \(\frac{1}{2} s\), then coasts at a steady angular velocity for another \(\frac{1}{2} s\) a. What is the speed of the dot at \(t=1.0 \mathrm{~s}\)? b. Through how many revolutions has the disk turned? Equation Transcription: Text Transcription: 8.0 cm 600 rad/s^2 1/2s 1/2s t=1.0 s

Problem 64P a. A turbine spinning with angular velocity ?0 rad/s comes to a halt in T seconds. Find an expression for the angle ?? through which the turbine turns while stopping. ________________ b. A turbine is spinning at 3800 rpm. Friction in the bearings is so low that it takes 10 min to coast to a slop. How many revolutions does the turbine make while stopping?

Problem 65P A high-speed drill rotating ccw at 2400 rpm comes to a halt in 2.5 s. a. What is the drill’s angular acceleration? ________________ b. How many revolutions does it make as it stops?

Problem 33E Section 4.7 Nonuniform Circular Motion and Angular Acceleration An electric fan goes from rest to 1800 rpm in 4.0 s. What is its angular acceleration?

Problem 34E Section 4.7 Nonuniform Circular Motion and Angular Acceleration A bicycle wheel is rotating at 50 rpm when the cyclist begins to pedal harder, giving the wheel a constant angular acceleration of 0.50rad/s2. a. What is the wheel’s angular velocity, in rpm, 10 s later? ________________ b. How many revolutions does the wheel make during this time?

Problem 35E Section 4.7 Nonuniform Circular Motion and Angular Acceleration A 3.0-cm-diameter crankshaft that is rotating at 2500 rpm comes to a halt in 1.5 s. a. What is the tangential acceleration of a point on the surface? ________________ b. How many revolutions does the crankshaft make as it stops?

A wheel initially rotating at \(60 \mathrm{rpm}\) experiences the angular acceleration shown in FigureP4.66 . What is the wheel's angular velocity, in rpm, at \(\mathrm{t}=3.0 \mathrm{~s}\)? Equation Transcription: Text Transcription: 60 rpm t=3.0 s

Problem 67P Your car tire is rotating at 3.5 rev/s when suddenly you press down hard on the accelerator. After traveling 200 m, the tire’s rotation has increased to 6.0 rev/s. What was the tire’s angular acceleration? Give your answer in rad/s2.

The angular velocity of a process control motor is \(\omega=\left(20-\frac{1}{2} t^{2}\right) \mathrm{rad} / \mathrm{s}\), where \(t\) is in seconds. a. At what time does the motor reverse direction? b. Through what angle does the motor turn between \(t=0 \mathrm{~s}\) and the instant at which it reverses direction? Equation Transcription: Text Transcription: omega=(20-1/2t^2)rad/s t t=0 s