Show that, in general, for any head-on

(I) Calculate the force exerted on a rocket when the propelling gases are being expelled at a rate of 1300 kg/s with a speed of 4.5 X 104 m/s.

(I) A constant friction force of 25 N acts on a 65-kg skier for 15 s. What is the skiers change in velocity?

(II) The momentum of a particle, in SI units, is given by p = 4.812\ - 8.0j - 8.9 tk. What is the force as a function of time?

The force on a particle of mass m is given by F = 26i - 1 2 12 j where F is in N and t in s. What will be the change in the particles momentum between t = 1.0 s and t = 2.0 s?

A 145-g baseball, moving along the x axis with speed 30.0 m/s, strikes a fence at a 45 angle and rebounds along the y axis with unchanged speed. Give its change in momentum using unit vector notation.

A 0.145-kg baseball pitched horizontally at 32.0 m/s strikes a bat and is popped straight up to a height of 36.5 m. If the contact time between bat and ball is 2.5 ms, calculate the average force between the ball and bat during contact.

A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s. To alter its course by 35.0, its rockets can be fired briefly in a direction perpendicular to its original motion. If the rocket gases are expelled at a speed of 1750 m/s, how much mass must be expelled?

Air in a 120-km/h wind strikes head-on the face of a building 45 m wide by 65 m high and is brought to rest. If air has a mass of 1.3 kg per cubic meter, determine the average force of the wind on the building.

A 7700-kg boxcar traveling 18 m/s strikes a second car. The two stick together and move off with a speed of 5.0 m/s. What is the mass of the second car?

A 9150-kg railroad car travels alone on a level frictionless track with a constant speed of 15.0 m/s. A 4350-kg load, initially at rest, is dropped onto the car. What will be the cars new speed?

An atomic nucleus at rest decays radioactively into an alpha particle and a smaller nucleus. What will be the speed of this recoiling nucleus if the speed of the alpha particle is 2.8 X 105m/s? Assume the recoiling nucleus has a mass 57 times greater than that of the alpha particle.

A 130-kg tackier moving at 2.5 m/s meets head-on (and tackles) an 82-kg halfback moving at 5.0 m/s. What will be their mutual speed immediately after the collision?

A child in a boat throws a 5.70-kg package out horizontally with a speed of 10.0 m/s, Fig. 9-37. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 24.0 kg and that of the boat is 35.0 kg. FIGURE 9-37 Problem 13.

An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction of its velocity, and the remaining nucleus slows to 350 m/s. If the alpha particle has a mass of 4.0 u and the original nucleus has a mass of 222 u, what speed does the alpha particle have when it is emitted?

An object at rest is suddenly broken apart into two fragments by an explosion. One fragment acquires twice the kinetic energy of the other. What is the ratio of their masses?

A 22-g bullet traveling 210 m/s penetrates a 2.0-kg block of wood and emerges going 150 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges?

A rocket of mass m traveling with speed v0 along the x axis suddenly shoots out fuel equal to one-third its mass, perpendicular to the x axis (along the y axis) with speed 2v0. Express the final velocity of the rocket in i, j, k notation.

The decay of a neutron into a proton, an electron, and a neutrino is an example of a three-particle decay process. Use the vector nature of momentum to show that if the neutron is initially at rest, the velocity vectors of the three must be coplanar (that is, all in the same plane). The result is not true for numbers greater than three.

A mass raA = 2.0 kg, moving with velocity \ A = (4.0i + 5.0j - 2.0k) m/s, collides with mass mB = 3.0 kg, which is initially at rest. Immediately after the collision, mass mA is observed traveling at velocity \ A = (-2.0i + 3.0k) m/s. Find the velocity of mass raB after the collision. Assume no outside force acts on the two masses during the collision.

A 925-kg two-stage rocket is traveling at a speed of 6.60 X 103 m/s away from Earth when a predesigned explosion separates the rocket into two sections of equal mass that then move with a speed of 2.80 X 103 m/s relative to each other along the original line of motion, (a) What is the speed and direction of each section (relative to Earth) after the explosion? (b) How much energy was supplied by the explosion? [Hint: What is the change in kinetic energy as a result of the explosion?]

A 224-kg projectile, fired with a speed of 116 m/s at a 60.0 angle, breaks into three pieces of equal mass at the highest point of its arc (where its velocity is horizontal). Two of the fragments move with the same speed right after the explosion as the entire projectile had just before the explosion; one of these moves vertically downward and the other horizontally. Determine (a) the velocity of the third fragment immediately after the explosion and (b) the energy released in the explosion.

A 0.145-kg baseball pitched at 35.0 m/s is hit on a horizontal line drive straight back at the pitcher at 56.0 m/s. If the contact time between bat and ball is 5.00 X 10-3 s, calculate the force (assumed to be constant) between the ball and bat.

A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact with the ball for 3.5 X 10 3 s. Find (a) the impulse imparted to the golf ball, and (b) the average force exerted on the ball by the golf club.

A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms. (a) What is the impulse given to the nail? (b) What is the average force acting on the nail?

A tennis ball of mass m = 0.060 kg and speed v = 25 m/s strikes a wall at a 45 angle and rebounds with the same speed at 45 (Fig. 9-38). What is the impulse (magnitude and direction) given to the ball?FIGURE 9-38 Problem 25.

A 130-kg astronaut (including space suit) acquires a speed of 2.50 m/s by pushing off with his legs from a 1700-kg space capsule, (a) What is the change in speed of the space capsule? (b) If the push lasts 0.500 s, what is the average force exerted by each on the other? As the reference frame, use the position of the capsule before the push, (c) What is the kinetic energy of each after the push?

Rain is falling at the rate of 5.0 cm/h and accumulates in a pan. If the raindrops hit at 8.0 m/s, estimate the force on the bottom of a 1.0 m2 pan due to the impacting rain which does not rebound. Water has a mass of 1.00 X 103 kg per m3.

Suppose the force acting on a tennis ball (mass 0.060 kg) points in the +x direction and is given by the graph of Fig. 9-39 as a function of time. Use graphical methods to estimate (a) the total impulse given the ball, and (b) the velocity of the ball after being struck, assuming the ball is being served so it is nearly at rest initially. FIGURE 9-39 Problem 28.

With what impulse does a 0.50-kg newspaper have to be thrown to give it a velocity of 3.0 m/s?

The force on a bullet is given by the formula F = [7 4 0 (2 .3 X 1 0 5 s_1) ? ] N over the time interval t = 0 to t = 3 .0 X 1 0 -3 s. (a) Plot a graph of F versus t for t = 0 to t = 3 .0 ms. (b) Use the graph to estimate the impulse given the bullet, (c) Determine the impulse by integration. (d) If the bullet achieves a speed of 2 6 0 m/s as a result of this impulse, given to it in the barrel of a gun, what must the bullets mass be? (e) What is the recoil speed of the 4.5-kg gun?

(a) A molecule of mass m and speed v strikes a wall at right angles and rebounds back with the same speed. If the collision time is A t, what is the average force on the wall during the collision? (b) If molecules, all of this type, strike the wall at intervals a time t apart (on the average) what is the average force on the wall averaged over a long time?

(a) Calculate the impulse experienced when a 65-kg person lands on firm ground after jumping from a height of 3.0 m. (ib) Estimate the average force exerted on the persons feet by the ground if the landing is stiff-legged, and again (c) with bent legs. With stiff legs, assume the body moves 1.0 cm during impact, and when the legs are bent, about 50 cm. [Hint: The average net force on her which is related to impulse, is the vector sum of gravity and the force exerted by the ground.]

A scale is adjusted so that when a large, shallow pan is placed on it, it reads zero. A water faucet at height h = 2.5 m above is turned on and water falls into the pan at a rate R = 0.14 kg/s. Determine (a) a formula for the scale reading as a function of time t and (b) the reading for t = 9.0 s. (c) Repeat (a) and (b), but replace the shallow pan with a tall, narrow cylindrical container of area A = 20 cm2 (the level rises in this case).

A 0.060-kg tennis ball, moving with a speed of 4.50 m/s, has a head-on collision with a 0.090-kg ball initially moving in the same direction at a speed of 3.00 m/s. Assuming a perfectly elastic collision, determine the speed and direction of each ball after the collision.

A 0.450-kg hockey puck, moving east with a speed of 4.80 m/s, has a head-on collision with a 0.900-kg puck initially at rest. Assuming a perfectly elastic collision, what will be the speed and direction of each object after the collision?

A 0.280-kg croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the original speed of the first ball. (a) What is the mass of the second ball? (b) What fraction of the original kinetic energy (AK /K ) gets transferred to the second ball?

A ball of mass 0.220 kg that is moving with a speed of 7.5 m/s collides head-on and elastically with another ball initially at rest. Immediately after the collision, the incoming ball bounces backward with a speed of 3.8 m/s. Calculate (a) the velocity of the target ball after the collision, and (b) the mass of the target ball.

A ball of mass m makes a head-on elastic collision with a second ball (at rest) and rebounds with a speed equal to 0.350 its original speed. What is the mass of the second ball?

Determine the fraction of kinetic energy lost by a neutron (mi = 1.01 u) when it collides head-on and elastically with a target particle at rest which is (a) }H (m = 1.01 u); (b) 2H (heavy hydrogen, m = 2.01 u); (c) x\C (m = 12.00 u); (d) 2jgPb (lead, m = 208 u).

Show that, in general, for any head-on one-dimensional elastic collision, the speeds after collision are 2 mA vb = vA and mA + raB fm B - mA + M \m---A-- -+7- -m---B ( mA mB\ ( + % 2 mB \ m A + m B / where vA and vB are the initial speeds of the two objects of mass mA and mB.

A 3.0-kg block slides along a frictionless tabletop at 8.0 m/s toward a second block (at rest) of mass 4.5 kg. A coil spring, which obeys Hookes law and has spring constant k = 850 N/m, is attached to the second block in such a way that it will be compressed when struck by the moving block, Fig. 9-40. (a) What will be the maximum compression of the spring? (b) What will be the final velocities of the blocks after the collision? (c) Is the collision elastic? Ignore the mass of the spring.FIGURE 9-40 Problem 41.

In a ballistic pendulum experiment, projectile 1 results in a maximum height h of the pendulum equal to 2.6 cm. A second projectile (of the same mass) causes the the pendulum to swing twice as high, h2 = 5.2 cm. The second projectile was how many times faster than the first?

(a) Derive a formula for the fraction of kinetic energy lost, A K /K, in terms of m and M for the ballistic pendulum collision of Example 9-11. (b) Evaluate for m = 16.0 g and M = 380 g.

A 28-g rifle bullet traveling 210 m/s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long string, which makes the pendulum swing upward in an arc. Determine the vertical and horizontal components of the pendulums maximum displacement.

An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 J is released in the explosion, how much kinetic energy does each piece acquire?

A 920-kg sports car collides into the rear end of a 2300-kg SUV stopped at a red light. The bumpers lock, the brakes are locked, and the two cars skid forward 2.8 m before stopping. The police officer, estimating the coefficient of kinetic friction between tires and road to be 0.80, calculates the speed of the sports car at impact. What was that speed?

You drop a 12-g ball from a height of 1.5 m and it only bounces back to a height of 0.75 m. What was the total impulse on the ball when it hit the floor? (Ignore air resistance).

Car A hits car B (initially at rest and of equal mass) from behind while going 35 m/s. Immediately after the collision, car B moves forward at 25 m /s and car A is at rest. What fraction of the initial kinetic energy is lost in the collision?

A measure of inelasticity in a head-on collision of two objects is the coefficient o f restitution, e, defined as e = Va ~ vb vb ~ v a where vA vB is the relative velocity of the two objects after the collision and vA is their relative velocity before it. (a) Show that e = 1 for a perfectly elastic collision, and e = 0 for a completely inelastic collision, (b) A simple method for measuring the coefficient of restitution for an object colliding with a very hard surface like steel is to drop the object onto a heavy steel plate, as shown in Fig. 9-41. Determine a formula for e in terms of the original height h and the maximum height h' reached after collision.FIGURE 9-41 Problem 49. Measurement of coefficient of restitution.

A pendulum consists of a mass M hanging at the bottom end of a massless rod of length , which has a frictionless pivot at its top end. A mass m, moving as shown N in Fig. 9-42 with velocity v, impacts M and becomes embedded. What iS the smallest value of v sufficient to cause the pendulum (with embedded mass m) to swing clear over the top of its arc?FIGURE 9-42 Problem 50.

A bullet of mass m = 0.0010 kg embeds itself in a wooden block with mass M = 0.999 kg, which then compresses a spring (k = 120 N/m) by a distance x = 0.050 m before coming to rest. The coefficient of kinetic friction between the block and table is fi = 0.50. (a) What is the initial speed of the bullet? (b) What fraction of the bullets initial kinetic energy is dissipated (in damage to the wooden block, rising temperature, etc.) in the collision between the bullet and the block?

A 144-g baseball moving 28.0 m/s strikes a stationary 5.25-kg brick resting on small rollers so it moves without significant friction. After hitting the brick, the baseball bounces straight back, and the brick moves forward at 1.10 m/s. (a) What is the baseballs speed after the collision? (b) Find the total kinetic energy before and after the collision.

A 6.0-kg object moving in the +x direction at 5.5 m/s collides head-on with an 8.0-kg object moving in the x direction at 4.0 m/s. Find the final velocity of each mass if: (a) the objects stick together; (b) the collision is elastic; (c) the 6.0-kg object is at rest after the collision; (d) the 8.0-kg object is at rest after the collision; (e) the 6.0-kg object has a velocity of 4.0 m/s in the x direction after the collision. Are the results in (c), (d), and (e) reasonable? Explain.

Billiard ball A of mass mA = 0.120 kg moving with speed vA = 2.80 m/s strikes ball B, initially at rest, of mass me = 0.140 kg. As a result of the collision, ball A is deflected off at an angle of 30.0 with a speed vA = 2.10 m/s. (a) Taking the x axis to be the original direction of motion of ball A, write down the equations expressing the conservation of momentum for the components in the x and y directions separately, (b) Solve these equations for the speed, vB, and angle, 0B, of ball B. Do not assume the collision is elastic.

A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of 9.6 X 10_23kg-m/s and 6.2 X 10_23kg-m/s, respectively. Determine the magnitude and the direc

Two billiard balls of equal mass move at right angles and meet at the origin of an xy coordinate system. Initially ball A is moving upward along the y axis at 2.0 m/s, and ball B is moving to the right along the x axis with speed 3.7 m/s. After the collision (assumed elastic), the second ball is moving along the positive y axis (Fig. 9-43). What is the final direction of ball A, and what are the speeds of the two balls?FIGURE 9-43 Problem 56. (Ball A after the collision is not shown.)

An atomic nucleus of mass m traveling with speed v collides elastically with a target particle of mass 2m (initially at rest) and is scattered at 90. (a) At what angle does the target particle move after the collision? (b) What are the final speeds of the two particles? (c) What fraction of the initial kinetic energy is transferred to the target particle?

A neutron collides elastically with a helium nucleus (at rest initially) whose mass is four times that of the neutron. The helium nucleus is observed to move off at an angle #He = 45. Determine the angle of the neutron, 0'n , and the speeds of the two particles, v n and ^He > after the collision. The neutrons initial speed is 6.2 X 105 m/s.

A neon atom (m = 20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 55.6 angle from its original direction and the unknown atom travels away at a -50.0 angle. What is the mass (in u) of the unknown atom? [Hint: You could use the law of sines.]

For an elastic collision between a projectile particle of mass mA and a target particle (at rest) of mass mB, show that the scattering angle, 0A , of the projectile (a) can take any value, 0 to 180, for mA < mB, but (b) has a maximum angle (f) given by cos2(f> = 1 (mB/mA)2 for mA > mB.

For an elastic collision between a projectile particle of mass mA and a target particle (at rest) of mass mB, show that the scattering angle, 0A , of the projectile (a) can take any value, 0 to 180, for mA < mB, but (b) has a maximum angle (f) given by cos2(f> = 1 (mB/mA)2 for mA > mB.

The cm of an empty 1250-kg car is 2.50 m behind the front of the car. How far from the front of the car will the cm be when two people sit in the front seat 2.80 m from the front of the car, and three people sit in the back seat 3.90 m from the front? Assume that each person has a mass of 70.0 kg.

The distance between a carbon atom (m = 12 u) and an oxygen atom (m = 16 u) in the CO molecule is 1.13 X 10- 10m. How far from the carbon atom is the center of mass of the molecule?

Three cubes, of side 0, 20, and 30, are placed next to one another (in contact) with their centers along a straight line as shown in Fig. 9-44. What is the position, along this line, of the cm of this r _ q system? Assume the cubes are made of the same uniform material. FIGURE 9-44 Problem 64.

A square uniform raft, 18 m by 18 m, of mass 6200 kg, is used as a ferryboat. If three cars, each of mass 1350 kg, occupy the NE, SE, and SW corners, determine the cm of the loaded ferryboat relative to the center of the raft.

A uniform circular plate of radius 2R has a circular hole of radius R cut out of it. The center C' of the smaller circle is a distance 0.80i? from the center C of the larger circle, Fig. 9-45. What is the position of the center of mass of the plate? [Hint: Try subtraction.] FIGURE 9-45 Problem 66.

A uniform thin wire is bent into a semicircle of radius r. Determine the coordinates of its center of mass with respect to an origin of coordinates at the center of the full circle.

Find the center of mass of the ammonia molecule. The chemical formula is NH3. The hydrogens are at the corners of an equilateral triangle (with sides 0.16 nm) that forms the base of a pyramid, with nitrogen at the apex (0.037 nm vertically above the plane of the triangle).

Determine the cm of a machine part that is a uniform cone of height h and radius R, Fig. 9-46. [Hint: Divide the cone into an infinite number ^ of disks of thickness dz, one of which is shown.l FIGURE 9-46 Problem 69.

Determine the cm of a uniform pyramid that has four triangular faces and a square base with equal sides all of length s. [Hint: See Problem 69.]

Determine the cm of a thin, uniform, semicircular plate.

Mass Ma = 35 kg and mass AfB = 25 kg. They have velocities (in m/s) vA = 12i - 16j and vB = -2 0 i + 14j. Determine the velocity of the center of mass of the system.

The masses of the Earth and Moon are 5.98 X 1024 kg and 7.35 X 1022 kg, respectively, and their centers are separated by 3.84 X 108m. (a) Where is the cm of this system located? (b) What can you say about the motion of the Earth-Moon system about the Sun, and of the Earth and Moon separately about the Sun?

A mallet consists of a uniform cylindrical head of mass 2.80 kg and a diameter 0.0800 m mounted on a uniform cylindrical handle of mass 0.500 kg and length 0.240 m, as shown in Fig. 9-47. If this mallet is tossed, spinning, into the air, how far above the bottom of the handle is the point that will follow a parabolic trajectory?FIGURE 9-47 Problem 74.

A 55-kg woman and a 72-kg man stand 10.0 m apart on frictionless ice. (a) How far from the woman is their cm? (b) If each holds one end of a rope, and the man pulls on the rope so that he moves 2.5 m, how far from the woman will he be now? (c) How far will the man have moved when he collides with the woman?

Suppose that in Example 9-18 (Fig. 9-32), ran = 3m i. (a) Where then would ran land? (b) What if mi = 3mn?

Two people, one of mass 85 kg and the other of mass 55 kg, sit in a rowboat of mass 78 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat, 3.0 m apart from each other, now exchange seats. How far and in what direction will the boat move?

A 280-kg flatcar 25 m long is moving with a speed of 6.0 m/s along horizontal frictionless rails. A 95-kg worker starts walking from one end of the car to the other in the direction of motion, with speed 2.0 m/s with respect to the car. In the time it takes for him to reach the other end, how far has the flatcar moved?

A huge balloon and its gondola, of mass M, are in the air and stationary with respect to the ground. A passenger, of mass m, then climbs out and slides down a rope with speed v, measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

A 3500-kg rocket is to be accelerated at 3.0 g at take-off from the Earth. If the gases can be ejected at a rate of 27 kg/s, what must be their exhaust speed?

Suppose the conveyor belt of Example 9-19 is retarded by a friction force of 150 N. Determine the required output power (hp) of the motor as a function of time from the moment gravel first starts falling (t = 0) until 3.0 s after the gravel begins to be dumped off the end of the 22-m-long conveyor belt.

The jet engine of an airplane takes in 120 kg of air per second, which is burned with 4.2 kg of fuel per second. The burned gases leave the plane at a speed of 550 m/s (relative to the plane). If the plane is traveling 270 m/s (600mi/h), determine: (a) the thrust due to ejected fuel; (b) the thrust due to accelerated air passing through the engine; and (c) the power (hp) delivered.

A rocket traveling 1850 m/s away from the Earth at an altitude of 6400 km fires its rockets, which eject gas at a speed of 1300 m/s (relative to the rocket). If the mass of the rocket at this moment is 25,000 kg and an acceleration of 1.5 m/s2 is desired, at what rate must the gases be ejected?

A sled filled with sand slides without friction down a 32 slope. Sand leaks out a hole in the sled at a rate of 2.0 kg/s. If the sled starts from rest with an initial total mass of 40.0 kg, how long does it take the sled to travel 120 m along the slope?

A novice pool player is faced with the corner pocket shot shown in Fig. 9-48. Relative dimensions are also shown. Should the player worry that this might be a scratch shot, in which the cue ball will also fall into a pocket? Give details. Assume equal mass balls and an elastic collision. FIGURE 9-48 Problem 85.

During a Chicago storm, winds can whip horizontally at speeds of 120 km/h. If the air strikes a person at the rate of 45 kg/s per square meter and is brought to rest, calculate the force of the wind on a person. Assume the person is 1.60 m high and 0.50 m wide. Compare to the typical maximum force of friction (/x ~ 1 .0) between the person and the ground, if the person has a mass of 75 kg.

A ball is dropped from a height of 1.50 m and rebounds to a height of 1.20 m. Approximately how many rebounds will the ball make before losing 90% of its energy?

In order to convert a tough split in bowling, it is necessary to strike the pin a glancing blow as shown in Fig. 9-49. Assume that the bowling ball, initially traveling at 13.0 m/s, has five times the mass of a pin and that the pin goes off at 75 from the original direction of the ball. Calculate the speed (a) of the pin and (b) of the ball just after collision, and (c) calculate the angle through which the ball was deflected. Assume the collision is elastic and ignore any spin of the ball. FIGURE 9-49 Problem 88.

A gun fires a bullet vertically into a 1.40-kg block of wood at rest on a thin horizontal sheet, Fig. 9-50. If the bullet has a mass of 24.0 g and a speed of 310 m/s, how high will the block rise into the air after the bullet becomes embedded in it? FIGURE 9-50 Problem 89.

A hockey puck of mass 4 m has been rigged to explode, as part of a practical joke. Initially the puck is at rest on a frictionless ice rink. Then it bursts into three pieces. One chunk, of mass m, slides across the ice at velocity vi. Another chunk, of mass 2m, slides across the ice at velocity 2vj. Determine the velocity of the third chunk.

For the completely inelastic collision of two railroad cars that we considered in Example 9-3, calculate how much of the initial kinetic energy is transformed to thermal or other forms of energy.

A 4800-kg open railroad car coasts along with a constant speed of 8.60 m/s on a level track. Snow begins to fall vertically and fills the car at a rate of 3.80 kg/min. Ignoring friction with the tracks, what is the speed of the car after 60.0 min? (See Section 9-2.)

Consider the railroad car of Problem 92, which is slowly filling with snow, (a) Determine the speed of the car as a function of time using Eqs. 9-19. (b ) What is the speed of the car after 60.0 min? Does this agree with the simpler calculation (Problem 92)?

Two blocks of mass raA and mB, resting on a frictionless table, are connected by a stretched spring and then released (Fig. 9-51). (a) Is there a net external force on the system? (b ) Determine the ratio of their speeds, v a / vq . (c) What is the ratio of their kinetic energies? (d) Describe the motion of the cm of this system. (e) How would the presence of friction alter the above results? FIGURE 9-51 Problem 94.

You have been hired as an expert witness in a court case involving an automobile accident. The accident involved car A of mass 1500 kg which crashed into stationary car B of mass 1100 kg. The driver of car A applied his brakes 15 m before he skidded and crashed into car B. After the collision, car A slid 18 m while car B slid 30 m. The coefficient of kinetic friction between the locked wheels and the road was measured to be 0.60. Show that the driver of car A was exceeding the 55-mi/h (90 km/h) speed limit before applying the brakes.

A meteor whose mass was about 2.0 X 108kg struck the Earth (raE = 6.0 X 1024kg) with a speed of about 25km/s and came to rest in the Earth, (a) What was the Earths recoil speed (relative to Earth at rest before the collision)? (b ) What fraction of the meteors kinetic energy was transformed to kinetic energy of the Earth? (c) By how much did the Earths kinetic energy change as a result of this collision?

Two astronauts, one of mass 65 kg and the other 85 kg, are initially at rest in outer space. They then push each other apart. How far apart are they when the lighter astronaut has moved 12 m?

A 22-g bullet strikes and becomes embedded in a 1.35-kg block of wood placed on a horizontal surface just in front of the gun. If the coefficient of kinetic friction between the block and the surface is 0.28, and the impact drives the block a distance of 8.5 m before it comes to rest, what was the muzzle speed of the bullet?

Two balls, of masses mA = 45 g and mB = 65 g, are suspended as shown in Fig. 9-52. The lighter ball is pulled away to a 66 angle with the vertical and released. (a) What is the velocity of the lighter ball before impact? (b) What is the velocity of each ball after the elastic collision? (c) What will be the maximum height of each ball after the elastic collision? FIGURE 9-52 Problem 99.

A block of mass ra = 2.20 kg slides down a 30.0 incline which is 3.60 m high. At the bottom, it strikes a block of mass M = 7.00 kg which is at rest on a horizontal surface, Fig. 9-53. (Assume a smooth transition at the bottom of the incline.) If the collision is elastic, and friction can be ignored, determine (a) the speeds of the two blocks after the collision, and (b) how far back up the incline the smaller mass will go. FIGURE 9-53 Problems 100 and 101.

In Problem 100 (Fig. 9-53), what is the upper limit on mass ra if it is to rebound from M, slide up the incline, stop, slide down the incline, and collide with M again?

After a completely inelastic collision between two objects of equal mass, each having initial speed, v, the two move off together with speed v/3. What was the angle between their initial directions?

A 0.25-kg skeet (clay target) is fired at an angle of 28 to the horizon with a speed of 25 m/s (Fig. 9-54). When it reaches the maximum height, h, it is hit from below by a 15-g pellet traveling vertically upward at a speed of 230 m/s. The pellet is embedded in the skeet. (a) How much higher, h', did the skeet go up? (b) How much extra distance, Ax, does the skeet travel because of the collision?FIGURE 9-54 Problem 103.

A massless spring with spring constant k is placed between a block of mass ra and a block of mass 3 ra. Initially the blocks are at rest on a frictionless surface and they are held together so that the spring between them is compressed by an amount D from its equilibrium length. The blocks are then released and the spring pushes them off in opposite directions. Find the speeds of the two blocks when they detach from the spring.

The gravitational slingshot effect. Figure 9-55 shows the planet Saturn moving in the negative x direction at its orbital speed (with respect to the Sun) of 9.6 km/s. The mass of Saturn is 5.69 X 1026 kg. A spacecraft with mass 825 kg approaches Saturn. When far from Saturn, it moves in the +x direction at 10.4 km/s. The gravitational attraction of Saturn (a conservative force) acting on the spacecraft causes it to swing around the planet (orbit shown as dashed line) and head off in the opposite direction. Estimate the final speed of the spacecraft after it is far enough away to be considered free of Saturns gravitational pull.

Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. 9-56). Car A has a mass of 450 kg and car B 490 kg, owing to differences in passenger mass. If car A approaches at 4.50 m/s and car B is moving at 3.70 m/s, calculate (a) their velocities after the collision, and (b) the change mA = in momentum of 450 kg each. FIGURE 9-56 U) Problem 106: (a) before collision, (b) after collision.

In a physics lab, a cube slides down frictionless incline as shown in Fig. 9-57 and elastically strikes another cube at the bottom that is only one-half its mass. If the incline is 35 cm high and the table is 95 cm off the floor, where does each cube land? [Hint: Both leave the incline moving horizontally.] FIGURE 9-57 Problem 107.

The space shuttle launches an 850-kg satellite by ejecting it from the cargo bay. The ejection mechanism is activated and is in contact with the satellite for 4.0 s to give it a velocity of 0.30 m/s in the z-direction relative to the shuttle. The mass of the shuttle is 92,000 kg. (a) Determine the component of velocity Vf of the shuttle in the minus z-direction resulting from the ejection, (b) Find the average force that the shuttle exerts on the satellite during the ejection.

You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing them into fixed, massive barriers at 45 km/h. A new model of mass 1500 kg takes 0.15 s from the time of impact until it is brought to rest, (a) Calculate the average force exerted on the car by the barrier, (b) Calculate the average deceleration of the car.

Astronomers estimate that a 2.0-km-wide asteroid collides with the Earth once every million years. The collision could pose a threat to life on Earth, (a) Assume a spherical asteroid has a mass of 3200 kg for each cubic meter of volume and moves toward the Earth at 15km/s. How much destructive energy could be released when it embeds itself in the Earth? (b) For comparison, a nuclear bomb could release about 4.0 X 1016J. How many such bombs would have to explode simultaneously to release the destructive energy of the asteroid collision with the Earth?

An astronaut of mass 210 kg including his suit and jet pack wants to acquire a velocity of 2.0 m/s to move back toward his space shuttle. Assuming the jet pack can eject gas with a velocity of 35 m/s, what mass of gas will need to be ejected?

An extrasolar planet can be detected by observing the wobble it produces on the star around which it revolves. Suppose an extrasolar planet of mass raB revolves around its star of mass mA. If no external force acts on this simple two-object system, then its cm is stationary. Assume mA and raB are in circular orbits with radii rA and rB about the systems cm. (a) Show that raB rA mA rB- (b) Now consider a Sun-like star and a single planet with the same characteristics as Jupiter. That is, mB = 1.0 X 10_3mA and the planet has an orbital radius of 8.0 X 1011 m. Determine the radius rA of the stars orbit about the systems cm. (c) When viewed from Earth, the distant system appears to wobble over a distance of 2 rA . If astronomers are able to detect angular displacements 6 of about 1 milliarcsec (1 arcsec = ^ of a degree), from what distance d (in light-years) can the stars wobble be detected (l ly = 9.46 X 1015 m)? (d) The star nearest to our Sun is about 4 ly away. Assuming stars are uniformly distributed throughout our region of the Milky Way Galaxy, about how many stars can this technique be applied to in the search for extrasolar planetary systems?

Suppose two asteroids strike head on. Asteroid A (mA = 7.5 X 1012kg) has velocity 3.3km/s before the collision, and asteroid B (mB = 1.45 X 1013kg) has velocity 1.4 km/s before the collision in the opposite direction. If the asteroids stick together, what is the velocity (magnitude and direction) of the new asteroid after the collision?

(Ill) A particle of mass raA traveling with speed v A collides elastically head-on with a stationary particle of smaller mass mB. (a) Show that the speed of mB after the collision is , _ 2va Vb 1 + mB/m A (b) Consider now a third particle of mass mc at rest between mA and mB so that mA first collides head on with mc and then mc collides head on with mB. Both collisions are elastic. Show that in this case, , _ _______mc mA_______ VB VA(mC + + mc) (c) From the result of part (b), show that for maximum v'B , mc = \Jm AmB. (d) Assume mB = 2.0 kg, mA = 18.0 kg and vA = 2.0 m/s. Use a spreadsheet to calculate and graph the values of v'B from rac = 0.0 kg to mc = 50.0 kg in steps of 1.0 kg. For what value of mc is the value of v B maximum? Does your numerical result agree with your result in part (c)?