a. With what minimum speed must you toss a 100 g ball

Problem 3E At what speed does a 1000 kg compact car have the same kinetic energy as a 20,000 kg truck going 25 km/h?

Calculate the ratio of the mean cube molar mass to the mean square molar mass in terms of (a) the fraction p, (b) the chain length.

Problem 4CQ Particle A has half the mass and eight times the kinetic energy of particle B. What is the speed ratio vA/vB?

Problem 4E a. What is the kinetic energy of a 1500 kg car traveling at a speed of 30 m/s (? 65 mph)? ________________ b. From what height would the car have to be dropped to have this same amount of kinetic energy just before impact? ________________ c. Does your answer to part b depend on the car’s mass?

Problem 5CQ A roller-coaster car rolls down a frictionless track, reaching speed v0 at the bottom. If you want the car to go twice as fast at the bottom, by what factor must you increase the height of the track? Explain.

The three balls in FIGURE Q10.6, which have equal masses, are fired with equal speeds from the same height above the ground. Rank in order, from largest to smallest, their speeds \(v_{\mathrm{a}}, v_{\mathrm{b}}\), and \(v_{\mathrm{c}}\) as they hit the ground. Explain. ________________ Equation Transcription: Text Transcription: v_a, v_b v_c

Problem 6E a. With what minimum speed must you toss a 100 g ball straight up to just barely hit the 10-m-high ceiling of the gymnasium if you release the ball 1.5 m above the floor? Solve this problem using energy. b. With what speed does the ball hit the floor?

Problem 9E What minimum speed does a 100 g puck need to make it to the top of a frictionless ramp that is 3.0 m long and inclined at 20°?

Problem 8CQ A spring has an unstretched length of 10 cm. It exerts a restoring force F when stretched to a length of 11 cm. a. For what length of the spring is its restoring force 3F? ________________ b. At what compressed length is the restoring force 2F?

Problem 7E A mother has four times the mass of her young son. Both are running with the same kinetic energy. What is the ratio vson/vmillher of their speeds?

The three balls in FIGURE Q10.7, which have equal masses, are fired with equal speeds at the angles shown. Rank in order, from largest to smallest, their speeds \(v_{a}, v_{b}\), and \(v_{c}\) as they cross the dashed horizontal line. Explain. (All three are fired with sufficient speed to reach the line.) ________________ Equation Transcription: Text Transcription: v_a, v_b v_c

Problem 9CQ The left end of a spring is attached to a wall. When Bob pulls on the right end with a 200 N force, he stretches the spring by 20 cm. The same spring is then used for a tug-of-war between Bob and Carlos. Each pulls on his end of the spring with a 200 N force. How far does the spring stretch? Explain.

Problem 8E A 55 kg skateboarder wants to just make it to the upper edge of a “quarter pipe,” a track that is one-quarter of a circle with a radius of 3.0 m. What speed does he need at the bottom?

Rank in order, from most to least, the elastic potential energy \(\left(U_{\mathrm{s}}\right)_{\mathrm{a}}\) to \(\left(U_{\mathrm{s}}\right)_{\mathrm{d}}\) stored in the springs of FIGURE Q10.10. Explain. ________________ Equation Transcription: Text Transcription: (U_s)_a (U_s)_d

Problem 11CQ A spring is compressed 1.0 cm. How far must you compress a spring with twice the spring constant to store the same amount of energy?

Problem 10E A pendulum is made by tying a 500 g ball to a 75-cm-long string. The pendulum is pulled 30° to one side, then released. a. What is the ball’s speed at the lowest point of its trajectory? ________________ b. To what angle does the pendulum swing on the other side?

A \(1500 kg\) car traveling at \(10 m/s\) suddenly runs out of gas while approaching the valley shown in Figure EX10.12. The alert driver immediately puts the car in neutral so that it will roll. What will be the car’s speed as it coasts into the gas station on the other side of the valley? ________________ Equation Transcription: Text Transcription: 1500 kg 10 m/s

Problem 11E A 20 kg child is on a swing that hangs from 3.0-m-long chains. What is her maximum speed if she swings out to a 45° angle?

Problem 12CQ A spring gun shoots out a plastic ball at speed v0. The spring is then compressed twice the distance it was on the first shot. By what factor is the ball’s speed increased? Explain.

A particle with the potential energy shown in Figure Q10.13 is moving to the right at \(x = 5 m\) with total energy \(E\). a. At what value or values of \(x\) is this particle’s speed a maximum? b. Does this particle have a turning point or points in the range of \(x\) covered by the graph? If so, where? c. If \(E\) is changed appropriately, could the particle remain at rest at any point or points in the range of \(x\) covered by the graph? If so, where? ________________ Equation Transcription: Text Transcription: x = 5 E x x E x

A \(5.0 kg\) mass hanging from a spring scale is slowly lowered onto a vertical spring, as shown in Figure EX10.17. The scale reads in newtons. a. What does the spring scale read just before the mass touches the lower spring? b. The scale reads 20 N when the lower spring has been compressed by \(2.0 cm\). What is the value of the spring constant for the lower spring? c. At what compression length will the scale read zero? ________________ Equation Transcription: Text Transcription: 5.0 kg 2.0 cm

Problem 14CQ Two balls of clay of known masses hang from the ceiling on mass-less strings of equal length. They barely touch when both hang at rest. One ball is pulled back until its string is at 45°, then released. It swings down, collides with the second ball, and they stick together. To determine the angle to which the balls swing on the opposite side, would you invoke (a) conservation of momentum, (b) conservation of mechanical energy, (c) both, (d) either but not both, or (e) these laws alone are not sufficient to find the angle? Explain.

Problem 13E You need to make a spring scale for measuring mass. You want each 1.0 cm length along the scale to correspond to a mass difference of 100 g. What should be the value of the spring constant?

Problem 14E A 10-cm-long spring is attached to the ceiling. When a 2.0 kg mass is hung from it, the spring stretches to a length of 15 cm. a. What is the spring constant k? ________________ b. How long is the spring when a 3.0 kg mass is suspended from it?

A \(60 \mathrm{~kg}\) student is standing atop a spring in an elevator as it accelerates upward at \(3.0 \mathrm{~m} / \mathrm{s}^{2}\). The spring constant is 2500 N/m. By how much is the spring compressed? ________________ Equation Transcription: Text Transcription: 60 kg 3.0 m/s^2

Problem 16E A spring banging from the ceiling has equilibrium length L0. Hanging mass m from the spring stretches its length to L1. Find an expression for the spring’s length L3 when mass 3m hangs from it.

Problem 18E How far must you stretch a spring with k = 1000 N/m to store 200 J of energy?

A stretched spring stores \(2.0 \mathrm{~J}\) of energy. How much energy will be stored if the spring is stretched three times as far?

Problem 20E A student places her 500 g physics book on a frictionless table. She pushes the book against a spring, compressing the spring by 4.0 cm, then releases the book. What is the book’s speed as it slides away? The spring constant is 1250 N/m.

Problem 21E A block sliding along a horizontal frictionless surface with speed v collides with a spring and compresses it by 2.0 cm. What will be the compression if the same block collides with the spring at a speed of 2v?

Problem 23E The desperate contestants on a TV survival show are very hungry. The only food they can see is some fruit hanging on a branch high in a tree. Fortunately, they have a spring they can use to launch a rock. The spring constant is 1000 N/m, and they can compress the spring a maximum of 30 cm. All the rocks on the island seem to have a mass of 400 g. a. With what speed does the rock leave the spring? ________________ b. If the fruit hangs 15 m above the ground, will they feast or go hungry?

Problem 22E 10 kg runaway grocery cart runs into a spring, attached to a wall, with spring constant 250 N/m and compresses it by 60 cm. What was the speed of the cart just before it hit the spring?

Problem 24E As a 15,000 kg jet lands on an aircraft carrier, its tail hook snags a cable to slow it down. The cable is attached to a spring with spring constant 60,000 N/m. If the spring stretches 30 m to stop the plane, what was the plane’s landing speed?

Figure EX10.25 is the potential-energy diagram for a \(20 g\) particle that is released from rest at \(x = 1.0 m\). a. Will the particle move to the right or to the left? How can you tell? b. What is the particle’s maximum speed? At what position does it have this speed? c. Where are the turning points of the motion? ________________ Equation Transcription: Text Transcription: 20 g x = 1.0 m

a. In Figure EX10.27, what minimum speed does a \(100 g\) particle need at point A to reach point B? b. What minimum speed does a \(100 g\) particle need at point B to reach point A? ________________ Equation Transcription: Text Transcription: 100 g 100 g

Figure EX10.26 is the potential energy diagram for a \(500 g\) particle that is released from rest at A. What are the particle’s speeds at B, C, and D? ________________ Equation Transcription: Text Transcription: 500 g

In Figure EX10.28, what is the maximum speed of a \(2.0 g\) particle that oscillates between \(x = 2.0 mm\) and \(x = 8.0 mm\)? ________________ Equation Transcription: Text Transcription: 2.0 g x = 2.0 mm x = 8.0 mm

Problem 29E A 50 g marble moving at 2.0 m/s strikes a 20 g marble at rest. What is the speed of each marble immediately after the collision? Assume the collision is perfectly elastic and the marbles collide head-on.

Problem 30E A proton is traveling to the right at 2.0 × 107 m/s. It has a head-on perfectly elastic collision with a carbon atom. The mass of the carbon atom is 12 times the mass of the proton. What are the speed and direction of each after the collision?

Problem 31E Ball 1, with a mass of 100 g and traveling at 10 m/s, collides head-on with ball 2, which has a mass of 300 g and is initially at rest. What are the final velocities of each ball if the collision is (a) perfectly elastic? (b) perfectly inelastic?

Problem 32E A 50 g ball of clay traveling at speed v0 hits and sticks to a 1.0 kg brick sitting at rest on a frictionless surface. a. What is the speed of the brick after the collision? ________________ b. What percentage of die mechanical energy is lost in this collision?

Problem 33P The maximum energy a bone can absorb without breaking is surprisingly small. Experimental data show that the leg bones of a healthy, 60 kg human can absorb about 200 J. a. From what maximum height could a 60 kg person jump and land rigidly upright on both feet without breaking bis legs? Assume that all energy is absorbed by the leg bones in a rigid landing. ________________ b. People jump safely from much greater heights than this. Explain how this is possible.

Problem 34P You’re driving at 35 km/h when the road suddenly descends 15 m into a valley. You take your foot off the accelerator and coast down the hill. Just as you reach the bottom you see the policeman hiding behind the speed limit sign that reads “70 km/h.” Are you going to get a speeding ticket?

Problem 35P A cannon tilted up at a 30° angle fires a cannon ball at 80 m/s from atop a 10-m-high fortress wall. What is the ball’s impact speed on the ground below? Ignore air resistance.

You have a ball of unknown mass, a spring with spring constant \(950 \mathrm{~N} / \mathrm{m}\), and a meter stick. You use various compressions of the spring to launch the ball vertically, then use the meter stick to measure the ball's maximum height above the launch point. Your data are as follows: Compression (cm) Height (cm) 2.0 32 3.0 65 4.0 115 5.0 189 Use an appropriate graph of the data to determine the ball’s mass.

Problem 37P A very slippery ice cube slides in a vertical plane around the inside of a smooth, 20-cm-diameter horizontal pipe. The ice cube’s speed at the bottom of the circle is 3.0 m/s. a. What is the ice cube’s speed at the top? ________________ b. Find an algebraic expression for the ice cube’s speed when it is at angle 9, where the angle is measured counterclockwise from the bottom of the circle. Your expression should give 3.0 m/s for ? = 0° and your answer to part a for ? = 180°.

A \(50 g\) rock is placed in a slingshot and the rubber band is stretched. The force of the rubber band on the rock is shown by the graph in Figure P10.38. a. Is the rubber band stretched to the right or to the left? How can you tell? b. Does this rubber band obey Hooke’s law? Explain. c. What is the rubber band’s spring constant \(k\)? d. The rubber band is stretched \(30\) cm and then released. What is the speed of the rock? ________________ Equation Transcription: Text Transcription: 50 g k 30 cm

The spring in FIGURE P10.40a is compressed by \(\Delta x\). It launches the block across a frictionless surface with speed \(v_{0}\). The two springs in FIGURE P10.40b are identical to the spring of Figure P10.40a. They are compressed by the same \(\Delta x\) and used to launch the same block. What is the block's speed now? ________________ Equation Transcription: Text Transcription: Delta x v_0 Delta x

Problem 39P The elastic energy stored in your tendons can contribute up to 35% of your energy needs when running. Sports scientists find that (on average) the knee extensor tendons in sprinters stretch 41 mm while those of nonathletes stretch only 33 mm. The spring constant of the tendon is the same for both groups, 33 N/mm. What is the difference in maximum stored energy between the sprinters and the nonathletes?

The spring in Figure P10.41a is compressed by \(\Delta x\). It launches the block across a frictionless surface with speed \(v_{0}\). The two springs in Figure P10.41b are identical to the spring of Figure P10.41a. They are compressed the same total \(\Delta x\) and used to launch the same block. What is the block’s speed now? ________________ Equation Transcription: Text Transcription: Delta x v_0 Delta x

Problem 42P a. A block of mass m can slide up and down a frictionless slope tilted at angle 6. The block is pressed against a spring at the bottom of the slope, compressing the spring (with spring constant k)by ?x, then released. Find an expression for the block’s maximum height h above its starting point. ________________ b. A 50 g ice cube can slide up and down a frictionless 30° slope. At the bottom, a spring with spring constant 25 N/m is compressed 10 cm and used to launch the ice cube up the slope. How high does it go?

Problem 45P A 1000 kg safe is 2.0 m above a heavy-duty spring when the rope holding the safe breaks. The safe hits the spring and compresses it 50 cm. What is the spring constant of the spring?

Problem 44P A 100 g granite cube slides down a 40° frictionless ramp. At the bottom, just as it exits onto a horizontal table, it collides with a 200 g steel cube at rest. How high above the table should the granite cube be released to give the steel cube a speed of 150 cm/s?

A package of mas \(m\) is released from rest at a warehouse loading dock and slides down the 3.0-m-high, frictionless chute of Figure P10.43 to a waiting truck. Unfortunately, the truck driver went on a break without having removed the previous package, of mass \(2m\) from the bottom of the chute. a. Suppose the packages stick together. What is their common speed after the collision? b. Suppose the collision between the packages is perfectly elastic. To what height does the package of mass \(m\) rebound? ________________ Equation Transcription: Text Transcription: m 2m m

Problem 47P You have been hired to design a spring-launched roller coaster that will carry two passengers per car. The car goes up a 10-m-high hill, then descends 15 in to the track’s lowest point. You’ve determined that the spring can be compressed a maximum of 2.0 m and that a loaded car will have a maximum mass of 400 kg. For safety reasons, the spring constant should be 10% larger than the minimum needed for the car to just make it over the top. a. What spring constant should you specify? ________________ b. What is the maximum speed of a 350 kg car if the spring is compressed the full amount?

It’s been a great day of new, frictionless snow. Julie starts at the top of the \(60^{\circ}\) slope shown in Figure P10.48. At the bottom, a circular arc carries her through a \(90^{\circ}\) turn, and she then launches off a 3.0-m-high ramp. How far horizontally is her touchdown point from the end of the ramp? ________________ Equation Transcription: Text Transcription: 60^circ 90^circ

Problem 46P A vertical spring with k =490 N/m is standing on the ground. You are holding a 5.0 kg block just above the spring, not quite touching it. a. How far does the spring compress if you let go of the block suddenly? ________________ b. How far does the spring compress if you slowly lower the block to the point where you can remove your hand without disturbing it? ________________ c. Why are your two answers different?

Problem 49P A 100 g block on a frictionless table is firmly attached to one end of a spring with k = 20 N/m. The other end of the spring is anchored to the wall. A 20 g ball is thrown horizontally toward the block with a speed of 5.0 m/s. a. If the collision is perfectly elastic, what is the ball’s speed immediately after the collision? ________________ b. What is the maximum compression of the spring? ________________ c. Repeat parts a and b for the case of a perfectly inelastic collision.

You have been asked to design a "ballistic spring system" to measure the speed of bullets. A bullet of mass \(m\) is fired into a block of mass \(M\). The block, with the embedded bullet, then slides across a frictionless table and collides with a horizontal spring whose spring constant is \(k\). The opposite end of the spring is anchored to a wall. The spring's maximum compression \(d\) is measured. a. Find an expression for the bullet's speed \(v_{\mathrm{B}}\) in terms of \(m, M, k\), and \(d\). b. What was the speed of a \(5.0 \mathrm{~g}\) bullet if the block's mass is \(2.0 \mathrm{~kg}\) and if the spring, with \(k=50 \mathrm{~N} / \mathrm{m}\), was compressed by \(10 \mathrm{~cm}\) ? c. What fraction of the bullet's energy is "lost"? Where did it go? ________________ Equation Transcription: Text Transcription: m M k d v_{\mathrm{B}} m, M, k d 5.0 g 2.0 kg k = 50 N/m 10 cm

Problem 51P You have been asked to design a “ballistic spring system” to measure the speed of bullets. A spring whose spring constant is k is suspended from the ceiling. A block of mass M hangs from the spring. A bullet of mass m is fired vertically upward into the bottom of the block and stops in the block. The spring’s maximum compression d is measured. a. Find an expression for the bullet’s speed vB in terms of m, M, k, and d. ________________ b. What was the speed of a 10 g bullet if the block’s mass is 2.0 kg and if the spring, with k = 50 N/m, was compressed by 45 cm?

Problem 53P A block of mass m slides down a frictionless track, then around the inside of a circular loop-the-loop of radius R. From what minimum height h must the block start to make it around without falling off? Give your answer as a multiple of R.

In Figure P10.52, a block of mass \(m\) slides along a frictionless track with speed \(v_{m}\). It collides with a stationary block of mass \(M\). Find an expression for the minimum value of \(v_{m}\) that will allow the second block to circle the loop-the-loop without falling off if the collision is (a) perfectly inelastic or (b) perfectly elastic. ________________ Equation Transcription: Text Transcription: m v_m M v_m

A new event has been proposed for the Winter Olympics. As seen in Figure P10.54, an athlete will sprint 100 m, starting from rest, then leap onto a \(20 kg\) bobsled. The person and bobsled will then slide down a 50-m-long ice-covered ramp, sloped at \(20^{\circ}\), and into a spring with a carefully calibrated spring constant of 2000 N/m. The athlete who compresses the spring the farthest wins the gold medal. Lisa, whose mass is \(40 kg\), has been training for this event. She can reach a maximum speed of 12 m/s in the 100 m dash. a. How far will Lisa compress the spring? b. The Olympic committee has very exact specifications about the shape and angle of the ramp. Is this necessary? What factors about the ramp are important? ________________ Equation Transcription: Text Transcription: 20 kg 20^circ 40 kg

Problem 56P A 100 g ball moving to the right at 4.0 m/s collides head-on with a 200 g ball that is moving to the left at 3.0 m/s. a. If the collision is perfectly elastic, what are the speed and direction of each ball after the collision? ________________ b. If the collision is perfectly inelastic, what are the speed and direction of the combined balls after the collision?

Problem 55P A 20 g ball is fired horizontally with speed v0 toward a 100 g ball hanging motionless from a 1.0-m-long suing. The balls undergo a head-on, perfectly elastic collision, after which the 100 g ball swings out to a maximum angle ?inax = 50°. What was v0?

Problem 57P A 100 g ball moving to the right at 4.0 m/s catches up and collides with a 400 g ball that is moving to the right at 1.0 m/s. If the collision is perfectly elastic, what are the speed and direction of each ball after the collision?

Figure P10.58 shows the potential energy of a \(500 g\) particle as it moves along the \(x\)-axis. Suppose the particle’s mechanical energy is 12 J. a. Where are the particle’s turning points? b. What is the particle’s speed when it is at \(x = 6.0 m\)? c. What is the particle’s maximum speed? At what position or positions does this occur? d. Write a description of the motion of the particle as it moves from the left turning point to the right turning point. e. Suppose the particle’s energy is lowered to 4.0 J. Describe the possible motions. ________________ Equation Transcription: Text Transcription: 500 g x x = 6.0 m

Problem 59P A particle has potential energy U(x)= x + sin((2 rad/m)x) over the range 0m ? x ? ? m. a. Where are the equilibrium positions in this range? ________________ b. For each, is it a point of stable or unstable equilibrium?

Protons and neutrons (together called nucleons) are held together in the nucleus of an atom by a force called the strong force. At very small separations, the strong force between two nucleons is larger than the repulsive electrical force between two protons - hence its name. But the strong force quickly weakens as the distance between the protons increases. A well-established model for the potential energy of two nucleons interacting via the strong force is \(U=U_{0}\left[1-e^{-x / x_{0}}\right]\) where \(x\) is the distance between the centers of the two nucleons, \(x_{0}\) is a constant having the value \(x_{0}=2.0 \times 10^{-15} \mathrm{~m}\), and \(U_{0}=6.0 \times 10^{-11} \mathrm{~J}\) a. Calculate and draw an accurate potential-energy curve from \(x=0 \mathrm{~m}\) to \(x=10 \times 10^{-15} \mathrm{~m}\). Either use your calculator to compute the value at several points or use computer software. b. Quantum effects are essential for a proper understanding of how nucleons behave. Nonetheless, let us innocently consider two neutrons as if they were small, hard, electrically neutral spheres of mass \(1.67 \times 10^{-27} \mathrm{~kg}\) and diameter \(1.0 \times 10^{-15} \mathrm{~m}\). (We will consider neutrons rather than protons so as to avoid complications from the electric forces between protons.) You are going to hold two neutrons \(5.0 \times 10^{-15} \mathrm{~m}\) apart, measured between their centers, then release them. Draw the total energy line for this situation on your diagram of part a. c. What is the speed of each neutron as they crash together? Keep in mind that both neutrons are moving. ________________ Equation Transcription: Text Transcription: U = U_{0}[1-e^{-x / x_{0}}] x x_0 x_0 = 2.0 X 10^-15 m U_0 = 6.0 X 10^11 J x = 0 m x = 10 X 10^-15 m 1.67 X 10^-27 kg 1.0 X 10^-15 m 5.0 X 10^-15 m

Write a realistic problem for which the energy bar chart shown in Figure P10.62 correctly shows the energy at the beginning and end of the problem.

In Problems 63 through 66 you are given the equation used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation. b. Draw the before-and-after pictorial representation. c. Finish the solution of the problem. \(\begin{aligned} & \frac{1}{2}(0.20 \mathrm{~kg})(2.0 \mathrm{~m} / \mathrm{s})^{2}+\frac{1}{2} k(0 \mathrm{~m})^{2} \\=& \frac{1}{2}(0.20 \mathrm{~kg})(0 \mathrm{~m} / \mathrm{s})^{2}+\frac{1}{2} k(-0.15 \mathrm{~m})^{2} \end{aligned}\) ________________ Equation Transcription: Text Transcription: frac{1}{2}(0.20 kg)(2.0 m / s)^{2} + frac{1}{2} k(0 m)^2 = frac{1}{2}(0.20 kg)(0 m / s)^{2} + frac{1}{2} k(-0.15 m)^2

In Problems 63 through 66 you are given the equation used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation. b. Draw the before-and-after pictorial representation. c. Finish the solution of the problem. \(\frac{1}{2}(1500 \mathrm{~kg})(5.0 \mathrm{~m} / \mathrm{s})^{2}+(1500 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(10 \mathrm{~m})\) \(=\frac{1}{2}(1500 \mathrm{~kg})\left(v_{\mathrm{i}}\right)^{2}+(1500 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(0 \mathrm{~m})\) ________________ Equation Transcription: Text Transcription: frac{1}{2}(1500kg)(5.0 m / s)^{2} + (1500 kg) (9.80 m / s}^{2})(10 m) = frac{1}{2}(1500 kg})(v_{i}})^{2} + (1500 kg) (9.80 m / s}^{2})(0 m)

Problem 61P A 50 g air-track glider is repelled by a post fixed at one end of the track. It is hypothesized that the glider’s potential energy is U = c/x, where x is the distance from the post and c is an unknown constant. To test this hypothesis, you launch the glider with the same speed at various distances from the post and then use a motion detector to measure its speed when it is 1.0 m from the post. Your data are as follows: Initial distance (cm) Speed at 1.0 m (m/s) 2.0 1.40 4.0 0.98 6.0 0.79 8.0 0.68 a. Do the data support the hypothesis? To find out, you’ll need to compare the shape of an appropriate graph to a theoretical prediction. ________________ b. Find an experimental value for c. Don’t forget to determine the appropriate units. Hint: Both the slope and the y-intercept of the graph are important.

In Problems 63 through 66 you are given the equation used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation. b. Draw the before-and-after pictorial representation. c. Finish the solution of the problem. \((0.10 \mathrm{~kg}+0.20 \mathrm{~kg}) v_{1 x}=(0.10 \mathrm{~kg})(3.0 \mathrm{~m} / \mathrm{s})\) \(\begin{aligned}&\frac{1}{2}(0.30 \mathrm{~kg})(0 \mathrm{~m} /\mathrm{s})^{2}+\frac{1}{2}(3.0 \mathrm{~N} / \mathrm{m})\left(\Delta x_{2}\right)^{2} \\&=\frac{1}{2}(0.30\mathrm{~kg})\left(v_{1 x}\right)^{2}+\frac{1}{2}(3.0 \mathrm{~N} / \mathrm{m})(0 \mathrm{~m})^{2}\end{aligned}\) ________________ Equation Transcription: Text Transcription: 0.10 g + 0.20 kg) v_{1 x} = (0.10 kg)(3.0 m / s}) frac{1}{2}(0.30 kg})(0 m/s}^{2} + frac{1}{2}(3.0 N/m)(Delta x_{2})^{2} = frac{1}{2}(0.30 kg)(v_1x)^{2} + frac{1}{2}(3.0 N/m)(0 {~m})^2

Problem 67CP A massless pan hangs from a spring that is suspended from the ceiling. When empty, the pan is 50 cm below the ceiling. If a 100 g clay ball is placed gently on the pan, the pan hangs 60 cm below the ceiling. Suppose the clay ball is dropped from the ceiling onto an empty pan. What is the pan’s distance from the ceiling when the spring reaches its maximum length?

In Problems 63 through 66 you are given the equation used to solve a problem. For each of these, you are to a. Write a realistic problem for which this is the correct equation. b. Draw the before-and-after pictorial representation. c. Finish the solution of the problem. \(\frac{1}{2}(0.50 \mathrm{~kg})\left(v_{\mathrm{f}}\right)^{2}+(0.50 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)(0 \mathrm{~m})\) \(+\frac{1}{2}(400 \mathrm{~N} / \mathrm{m})(0 \mathrm{~m})^{2}=\frac{1}{2}(0.50 \mathrm{~kg})(0 \mathrm{~m} / \mathrm{s})^{2}\) \(+(0.50 \mathrm{~kg})\left(9.80 \mathrm{~m} / \mathrm{s}^{2}\right)\left((-0.10 \mathrm{~m}) \sin 30^{\circ}\right)\) \(+\frac{1}{2}(400 \mathrm{~N} / \mathrm{m})(-0.10 \mathrm{~m})^{2}\) ________________ Equation Transcription: Text Transcription: frac{1}{2}(0.50 kg) (v_{\mathrm{f}})^{2} + (0.50 kg)(9.80 m / s^{2})(0 m) +frac{1}{2}(400 N / m)(0 m)^{2} = frac{1}{2}(0.50 kg)(0 m / s)^2 +(0.50 kg})(9.80 m /{s}^{2})((-0.10 m) sin 30^circ) +frac{1}{2}(400N/m)(-0.10 m)^{2}

A pendulum is formed from a small ball of mass \(m\) on a string of length \(L\). As Figure CP10.68 shows, a peg is height \(h = L/3\) above the pendulum’s lowest point. From what minimum angle \(theta\0 must the pendulum be released in order for the ball to go over the top of the peg without the string going slack? ________________ Equation Transcription: Text Transcription: m L h = L/3 theta

It’s your birthday, and to celebrate you’re going to make your first bungee jump. You stand on a bridge 100 m above a raging river and attach a 30-m-long bungee cord to your harness. A bungee cord, for practical purposes, is just a long spring, and this cord has a spring constant of 40 N/m. Assume that your mass is 80 kg. After a long hesitation, you dive off the bridge. How far are you above the water when the cord reaches its maximum elongation?

A \(10 \ kg\) box slides \(4.0 \ m\) down the frictionless ramp shown in Figure CP10.71, then collides with a spring whose spring constant is \(250 \ N/m\). a. What is the maximum compression of the spring? b. At what compression of the spring does the box have its maximum speed? Equation Transcription: Text Transcription: 10 kg 4.0 m 250 N/m

Problem 69CP In a physics lab experiment, a compressed spring launches a 20 g metal ball at a 30° angle. Compressing the spring 20 cm causes the ball to hit the floor 1.5 m below the point at which it leaves the spring after traveling 5.0 m horizontally. What is the spring constant?

Problem 72CP Old naval ships fired 10 kg cannon balls from a 200 kg cannon. It was very important to stop the recoil of the cannon, since otherwise the heavy cannon would go careening across the deck of the ship. In one design, a large spring with spring constant 20,000 N/m was placed behind the cannon. The other end of the spring braced against a post that was firmly anchored to the ship’s frame. What was the speed of the cannon ball if the spring compressed 50 cm when the cannon was fired?

Problem 73CP A 2.0 kg cart has a spring with k = 5000 N/m attached to its front, parallel to the ground. This cart rolls at 4.0 m/s toward a stationary 1.0 kg cart. a. What is the maximum compression of the spring during the collision? ________________ b. What is the speed of each cart after the collision?

The air-track carts in Figure CP10.74 are sliding to the right at \(1.0 m/s\). The spring between them has a spring constant of 120 N/m and is compressed \(4.0 cm\). The carts slide past a flame that burns through the string holding them together. Afterward, what are the speed and direction of each cart? ________________ Equation Transcription: Text Transcription: 1.0 m/s 4.0 cm

Problem 75CP A 100 g steel ball and a 200 g steel ball each hang from 1.0-m-long strings. At rest, the balls hang side by side, barely touching. The 100 g ball is pulled to the left until the angle between its string and vertical is 45°. The 200 g ball is pulled to a 45° angle on the right. The balls are released so as to collide at the very bottom of their swings. To what angle does each ball rebound?

A sled starts from rest at the top of the frictionless, hemispherical, snow-covered hill shown in Figure CP10.76. a. Find an expression for the sled's speed when it is at angle \(\phi\). b. Use Newton's laws to find the maximum speed the sled can have at angle \(\phi\) without leaving the surface. c. At what angle \(\phi_{\max }\) does the sled "fly off" the hill? ________________ Equation Transcription: Text Transcription: phi phi phi_max

Problem 1CQ Upon what basic quantity does kinetic energy depend? Upon what basic quantity does potential energy depend?

Problem 1E Which has the larger kinetic energy, a 10 g bullet fired at 500 m/s or a 10 kg bowling ball sliding at 10 m/s?

Problem 2CQ Can kinetic energy ever be negative? Can gravitational potential energy ever be negative? For each, give a plausible reason for your answer without making use of any equations.

Problem 2E The lowest point in Death Valley is 85 m below sea level. The summit of nearby Mt. Whitney has an elevation of 4420 m. What is the change in potential energy of an energetic 65 kg hiker who makes it from the floor of Death Valley to the top of Mt. Whitney?

Problem 5E A boy reaches out of a window and tosses a ball straight up with a speed of 10 m/s. The ball is 20 m above the ground as he releases it. Use conservation of energy to find a. The ball’s maximum height above the ground. b. The ball’s speed as it passes the window on its way down. c. The speed of impact on the ground.