Chapter 8

BIO Changing Your Center of Mass. To keep the calculations fairly simple, but still reasonable, we shall model a human leg that is 92.0 \(\mathrm{cm}\) long (measured from the hip joint) by assuming that the upper leg and the lower leg (which includes the foot) have equal lengths and that each of them is uniform. For a 70.0 -kg person, the mass of the upper leg would be \(8.60 \mathrm{kg},\) while that of the lower leg (including the foot) would be 5.25 \(\mathrm{kg} .\) Find the location of the center of mass of this leg, relative to the hip joint, if it is (a) stretched out horizontally and (b) bent at the knee to form a right angle with the upper leg remaining horizontal.

A 70 -kg astronaut floating in space in a 110 -kg MMU (manned maneuvering unit) experiences an acceleration of 0.029 \(\mathrm{m} / \mathrm{s}^{2}\) when he fires one of the MMU's thrusters. (a) If the speed of the escaping \(\mathrm{N}_{2}\) gas relative to the astronaut is \(490 \mathrm{m} / \mathrm{s},\) how much gas is used by the thruster in 5.0 \(\mathrm{s} ?\) (b) What is the thrust of the thruster?

A small rocket burns 0.0500 \(\mathrm{kg}\) of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 \(\mathrm{m} / \mathrm{s}\) . (a) What is the thrust of the rocket? (b) Would the rocket operate in outer space where there is no atmosphere? If so, how would you steer it? Could you brake it?

A C6-5 model rocket engine has an impulse of 10.0 \(\mathrm{N} \cdot \mathrm{s}\) while burning 0.0125 \(\mathrm{kg}\) of propellant in 1.70 s. It has a maximum thrust of 13.3 \(\mathrm{N} .\) The initial mass of the engine plus propellant is 0.0258 \(\mathrm{kg} .\) (a) What fraction of the maximum thrust is the average thrust? (b) Calculate the relative speed of the exhaust gases, assuming it is constant. (c) Assuming that the relative speed of the exhaust gases is constant, find the final speed of the engine if it was attached to a very light frame and fired from rest in gravity-free outer space.

Obviously, we can make rockets to go very fast, but what is a reasonable top speed? Assume that a rocket is fired from rest at a space station in deep space, where gravity is negligible. (a) If the rocket ejects gas at a relative speed of 2000 \(\mathrm{m} / \mathrm{s}\) and you want the rocket's speed eventually to be \(1.00 \times 10^{-3} \mathrm{c}\) , where \(c\) is the speed of light, what fraction of the initial mass of the rocket and fuel is not fuel? (b) What is this fraction if the final speed is to be 3000 \(\mathrm{m} / \mathrm{s} ?\)

A single-stage rocket is fired from rest from a deep-space platform, where gravity is negligible. If the rocket burns its fuel in 50.0 s and the relative speed of the exhaust gas is \(v_{\text { ex }}=2100 \mathrm{m} / \mathrm{s}\) what must the mass ratio \(m_{0} / m\) be for a final speed \(v\) of 8.00 \(\mathrm{km} / \mathrm{s}\) (about equal to the orbital speed of an earth satellite)?

CP CALC A young girl with mass 40.0 \(\mathrm{kg}\) is sliding on a horizontal, frictionless surface with an initial momentum that is due east and that has magnitude 90.0 \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) Starting at \(t=0, \mathrm{a}\) net force with magnitude \(F=(8.20 \mathrm{N} / \mathrm{s}) t\) and direction due west is applied to the girl. (a) At what value of \(t\) does the girl have a westward momentum of magnitude 60.0 \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s} ?\) (b) How much work has been done on the girl by the force in the time interval from \(t=0\) to the time calculated in part (a)? (c) What is the magnitude of the acceleration of the girl at the time calculated in part (a)?

A steel ball with mass 40.0 \(\mathrm{g}\) is dropped from a height of 2.00 \(\mathrm{m}\) onto a horizontal steel slab. The ball rebounds to a height of 1.60 \(\mathrm{m} .\) (a) Calculate the impulse delivered to the ball during impact. (b) If the ball is in contact with the slab for 2.00 \(\mathrm{ms}\) , find the average force on the ball during impact.

In a volcanic eruption, a \(2400\)-kg boulder is thrown vertically upward into the air. At its highest point, it suddenly explodes (due to trapped gases) into two fragments, one being three times the mass of the other. The lighter fragment starts out with only horizontal velocity and lands 318 \(\mathrm{m}\) directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

Just before it is struck by a racket, a tennis ball weighing 0.560 \(\mathrm{N}\) has a velocity of \((20.0 \mathrm{m} / \mathrm{s}) \hat{\imath}-(4.0 \mathrm{m} / \mathrm{s}) \hat{\boldsymbol{J}}\) . During the 3.00 \(\mathrm{ms}\) that the racket and ball are in contact, the net force on the ball is constant and equal to \(-(380 \mathrm{N}) \hat{\boldsymbol{\imath}}+(110 \mathrm{N}) \hat{\boldsymbol{J}}\) . (a) What are the \(x\) - and \(y\) -components of the impulse of the net force applied to the ball? (b) What are the \(x\) - and \(y\) -components of the final velocity of the ball?

Three identical pucks on a horizontal air table have repelling magnets. They are held together and then released simultaneously. Each has the same speed at any instant. One puck moves due west. What is the direction of the velocity of each of the other two pucks?

A 1500 -kg blue convertible is traveling south, and a \(2000-\mathrm{kg}\) red SUV is traveling west. If the total momentum of the system consisting of the two cars is 7200 \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) directed at \(60.0^{\circ}\) west of south, what is the speed of each vehicle?

A railroad handcar is moving along straight, frictionless tracks with negligible air resistance. In the following cases, the car initially has a total mass (car and contents) of 200 \(\mathrm{kg}\) and is traveling east with a velocity of magnitude 5.00 \(\mathrm{m} / \mathrm{s} .\) Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks. (a) A \(25.0-\mathrm{kg}\) mass is thrown sideways out of the car with a velocity of magnitude 2.00 \(\mathrm{m} / \mathrm{s}\) relative to the car's initial velocity. (b) A \(25.0-\mathrm{kg}\) mass is thrown backward out of the car with a velocity of 5.00 \(\mathrm{m} / \mathrm{s}\) relative to the initial motion of the car. (c) A 25.0 -kg mass is thrown into the car with a velocity of 6.00 \(\mathrm{m} / \mathrm{s}\) relative to the ground and opposite in direction to the initial velocity of the car.

You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass \(80.0 \mathrm{kg},\) is given a push and slides eastward. Abigail, with mass \(50.0 \mathrm{kg},\) is sent sliding northward. They collide, and after the collision Sam is moving at \(37.0^{\circ}\) north of east with a speed of 6.00 \(\mathrm{m} / \mathrm{s}\) and Abigail is moving at \(23.0^{\circ}\) south of east with a speed of 9.00 \(\mathrm{m} / \mathrm{s}\) (a) What was the speed of each person before the collision? (b) By how much did the total kinetic energy of the two people decrease during the collision?

The nucleus of \(^{214} \mathrm{Po}\) decays radioactively by emitting an alpha particle (mass \(6.65 \times 10^{-27} \mathrm{kg} )\) with kinetic energy \(1.23 \times\) \(10^{-12} \mathrm{J},\) as measured in the laboratory reference frame. Assuming that the Po was initially at rest in this frame, find the recoil velocity the nucleus that remains after the decay.

At a classic auto show, a \(840-\mathrm{kg} 1955\) Nash Metropolitan motors by at 9.0 \(\mathrm{m} / \mathrm{s}\) , followed by a \(1620-\mathrm{kg} 1957\) Packard Clipper purring past at 5.0 \(\mathrm{m} / \mathrm{s}\) . (a) Which car has the greater kinetic energy? What is the ratio of the kinetic energy of the Nash to that of the Packard? (b) Which car has the greater magnitude of momentum? What is the ratio of the magnitude of momentum of the Nash to that of the Packard? (c) Let \(F_{\mathrm{N}}\) be the net force required to stop the Nash in time \(t,\) and let \(F_{\mathrm{P}}\) be the net force required to stop the Packard in the same time. Which is larger: \(F_{\mathrm{N}}\) or \(F_{\mathrm{P}} ?\) What is the ratio \(F_{\mathrm{N}} / F_{\mathrm{P}}\) of these two forces? (d) Now let \(F_{\mathrm{N}}\) be the net force required to stop the Nash in a distance \(d,\) and let \(F_{\mathrm{P}}\) be the net force required to stop the Packard in the same distance. Which is larger: \(F_{\mathrm{N}}\) or \(F_{\mathrm{P}} ?\) What is the ratio \(F_{\mathrm{N}} / F_{\mathrm{P}}\)?

An 8.00 -kg block of wood sits at the edge of a frictionless table, 2.20 \(\mathrm{m}\) above the floor. A \(0.500-\mathrm{kg}\) blob of clay slides along the length of the table with a speed of 24.0 \(\mathrm{m} / \mathrm{s}\), strikes the block of wood, and sticks to it. The combined object leaves the edge of the table and travels to the floor. What horizontal distance has the combined object traveled when it reaches the floor?

A small wooden block with mass 0.800 \(\mathrm{kg}\) is suspended from the lower end of a light cord that is 1.60 \(\mathrm{m}\) long. The block is initially at rest. A bullet with mass 12.0 \(\mathrm{g}\) is fired at the block with a horizontal velocity \(v_{0} .\) The bullet strikes the block and becomes embedded in it. After the collision the combined object swings on the end of the cord. When the block has risen a vertical height of \(0.800 \mathrm{m},\) the tension in the cord is 4.80 \(\mathrm{N} .\) What was the initial speed \(v_{0}\) of the bullet?

Combining Conservation Laws. A 5.00 -kg chunk of ice is sliding at 12.0 \(\mathrm{m} / \mathrm{s}\) on the floor of an ice-covered valley when it collides with and sticks to another 5.00 -kg chunk of ice that is initially at rest. Fig. \(P 8.79\) ). Since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go?

Automobile Accident Analysis. You are called as an expert witness to analyze the following auto accident: Car \(B,\) of mass \(1900 \mathrm{kg},\) was stopped at a red light when it was hit from behind by car \(A,\) of mass 1500 \(\mathrm{kg}\) . The cars locked bumpers during the collision and slid to a stop with brakes locked on all wheels. Measurements of the skid marks left by the tires showed them to be 7.15 \(\mathrm{m}\) long. The coefficient of kinetic friction between the tires and the road was 0.65 . (a) What was the speed of car A just before the collision? (b) If the speed limit was 35 \(\mathrm{mph}\) , was car \(A\) speeding, and if so, by how many miles per hour was it exceeding the speed limit?

A 0.150 -kg frame, when suspended from a coil spring, stretches the spring 0.070 m. A 0.200 -kg lump of putty is dropped from rest onto the frame from a height of 30.0 \(\mathrm{cm}\) (Fig. P8.82). Find the maximum distance the frame moves downward from its initial position.

A rifle bullet with mass 8.00 \(\mathrm{g}\) strikes and embeds itself in a block with mass 0.992 \(\mathrm{kg}\) that rests on a frictionless, horizontal surface and is attached to a coil spring (Fig. P8.83). The impact compresses the spring 15.0 \(\mathrm{cm} .\) Calibration of the spring shows that a force of 0.750 \(\mathrm{N}\) is required to compress the spring 0.250 \(\mathrm{cm} .\) (a) Find the magnitude of the block's velocity just after impact. (b) What was the initial speed of the bullet?

A Ricocheting Bullet. 0.100 -kg stone rests on a frictionless, horizontal surface. A bullet of mass 6.00 g, traveling horizontally at 350 \(\mathrm{m} / \mathrm{s}\) , strikes the stone and rebounds horizontally at right angles to its original direction with a speed of 250 \(\mathrm{m} / \mathrm{s}\) . (a) Compute the magnitude and direction of the velocity of the stone after it is struck. (b) Is the collision perfectly elastic?

A movie stuntman (mass 80.0 \(\mathrm{kg}\) ) stands on a window ledge 5.0 \(\mathrm{m}\) above the floor (Fig. P8.85). Grabbing a rope attached to a chandelier, he swings down to grapple with the movie's villain (mass 70.0 kg), who is standing directly under the chandelier. (Assume that the stuntman's center of mass moves downward 5.0 \(\mathrm{m} .\) He releases the rope just as he reaches the villain.) (a) With what speed do the entwined foes start to slide across the floor? (b) If the coefficient of kinetic friction of their bodies with the floor is \(\mu_{k}=0.250,\) how far do they slide?

A ball with mass \(M\) , moving horizontally at \(4.00 \mathrm{m} / \mathrm{s},\) collides elastically with a block with mass 3\(M\) that is initially hanging at rest from the ceiling on the end of a \(50.0-\) cm wire. Find the maximum angle through which the block swings after it is hit.

A \(20.00-\) -kg lead sphere is hanging from a hook by a thin wire 3.50 \(\mathrm{m}\) long and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00 -kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a complete circular loop after the collision?

An \(8.00-\mathrm{kg}\) ball, hanging from the ceiling by a light wire 135 \(\mathrm{cm}\) long, is struck in an elastic collision by a 2.00 -kg ball moving horizontally at 5.00 \(\mathrm{m} / \mathrm{s}\) just before the collision. Find the tension in the wire just after the collision.

A 7.0 -kg shell at rest explodes into two fragments, one with a mass of 2.0 \(\mathrm{kg}\) and the other with a mass of 5.0 \(\mathrm{kg}\) . If the heavier fragment gains 100 \(\mathrm{J}\) of kinetic energy from the explosion, how much kinetic energy does the lighter one gain?

A 4.00-g bullet, traveling horizontally with a velocity of magnitude \(400 \mathrm{m} / \mathrm{s},\) is fired into a wooden block with mass \(0.800 \mathrm{kg},\) initially at rest on a level surface. The bullet passes through the block and emerges with its speed reduced to 190 \(\mathrm{m} / \mathrm{s}\) . The block slides a distance of 45.0 \(\mathrm{cm}\) along the surface from its initial position. (a) What is the coefficient of kinetic friction between block and surface? (b) What is the decrease in kinetic energy of the bullet? (c) What is the kinetic energy of the block at the instant after the bullet passes through it?

A 5.00 -g bullet is shot through a 1.00 -kg wood block suspended on a string 2.00 \(\mathrm{m}\) long. The center of mass of the block rises a distance of 0.38 \(\mathrm{cm} .\) Find the speed of the bullet as it emerges from the block if its initial speed is 450 \(\mathrm{m} / \mathrm{s}\) .

A neutron with mass \(m\) makes a head-on, elastic collision with a nucleus of mass \(M,\) which is initially at rest. (a) Show that if the neutron's initial kinetic energy is \(K_{0}\) , the kinetic energy that it loses during the collision is 4\(m M K_{0} /(M+m)^{2}\) . (b) For what value of \(M\) does the incident neutron lose the most energy? (c) When \(M\) has the value calculated in part (b), what is the speed of the neutron after the collision?

Energy Sharing in Elastic Collisions. A stationary object with mass \(m_{B}\) is struck head-on by an object with mass \(m_{A}\) that is moving initially at speed \(v_{0} .\) (a) If the collision is elastic, what percentage of the original energy does each object have after the collision? (b) What does your answer in part (a) give for the special cases (i) \(m_{A}=m_{B}\) and (ii) \(m_{A}=5 m_{B} ?(\mathrm{c})\) For what values, if any, of the mass ratio \(m_{A} / m_{B}\) is the original kinetic energy shared equally by the two objects after the collision?

In a shipping company distribution center, an open cart of mass 50.0 \(\mathrm{kg}\) is rolling to the left at a speed of 5.00 \(\mathrm{m} / \mathrm{s}\) (Fig. P8.95). You can ignore friction between the cart and the floor. A 15.0 -kg package slides down a chute that is inclined at \(37^{\circ}\) from the horizontal and leaves the end of the chute with a speed of 3.00 \(\mathrm{m} / \mathrm{s}\) . The package lands in the cart and they roll off together. If the lower end roll off together. If the lower end of the chute is a vertical distance of 4.00 \(\mathrm{m}\) above the bottom of the cart, what are (a) the speed of the package just before it lands in the cart and (b) the final speed of the cart?

A blue puck with mass \(0.0400 \mathrm{kg},\) sliding with a velocity of magnitude 0.200 \(\mathrm{m} / \mathrm{s}\) on a frictionless, horizontal air table, makes a perfectly elastic, head-on collision with a red puck with mass \(m,\) initially at rest. After the collision, the velocity of the blue puck is 0.050 \(\mathrm{m} / \mathrm{s}\) in the same direction as its initial velocity. Find (a) the velocity (magnitude and direction) of the red puck after the collision and (b) the mass \(m\) of the red puck.

Jack and Jill are standing on a crate at rest on the frictionless, horizontal surface of a frozen pond. Jack has mass 75.0 kg, Jill has mass \(45.0 \mathrm{kg},\) and the crate has mass 15.0 \(\mathrm{kg}\) . They remember that they must fetch a pail of water, so each jumps horizontally from the top of the crate. Just after each jumps, that person is moving away from the crate. with a speed of 4.00 \(\mathrm{m} / \mathrm{s}\) relative to the crate. (a) What is the final speed of the crate if both Jack and Jill jump simultaneously and in the same direction? (Hint: Use an inertial coordinate system attached to the ground.) (b) What is the final speed of the crate if Jack jumps first and then a few seconds later Jill jumps in the same direction? (c) What is the final speed of the crate if Jill jumps first and then Jack, again in the same direction?

Suppose you hold a small ball in contact with, and directly over, the center of a large ball. If you then drop the small ball a short time after dropping the large ball, the small ball rebounds with surprising speed. To show the extreme case, ignore air resistance and suppose the large ball makes an elastic collision with the floor and then rebounds to make an elastic collision with the still-descending small. Just before the collision between the two balls, the large ball is moving upward with velocity \(\vec{\boldsymbol{v}}\) and the small ball has velocity \(-\vec{\boldsymbol{v}}\) . (Do you see why? \()\) Assume the large ball has a much greater mass than the small ball. (a) What is the velocity of the small ball immediately after its collision with the large ball? (b) From the answer to part (a), what is the ratio of the small ball's rebound distance to the distance it fell before the collision?

Hockey puck \(B\) rests on a smooth ice surface and is struck by a second puck \(A,\) which has the same mass. Puck \(A\) is initially traveling at 15.0 \(\mathrm{m} / \mathrm{s}\) and is deflected \(25.0^{\circ}\) from its initial direction. Assume that the collision is perfectly elastic. Find the final speed of each puck and the direction of \(B\) 's velocity after the collision.

Energy Sharing. An object with mass \(m,\) initially at rest, explodes into two fragments, one with mass \(m_{A}\) and the other with mass \(m_{B},\) where \(m_{A}+m_{B}=m .\) (a) If energy \(Q\) is released in the explosion, how much kinetic energy does each fragment have immediately after the explosion? (b) What percentage of the total energy released does each fragment get when one fragment has four times the mass of the other?

\(\mathrm{A}^{232} \mathrm{Th}\) (thorium) nucleus at rest decays to a \(^{228} \mathrm{Ra}\) (radium) nucleus with the emission of an alpha particle. The total kinetic energy of the decay fragments is \(6.54 \times 10^{-13} \mathrm{J}\) . An alpha particle has 1.76\(\%\) of the mass of a \(^{228} \mathrm{Ra}\) nucleus. Calculate the kinetic energy of (a) the recoiling 228 nucleus and (b) the alpha particle.

Antineutrino. In beta decay, a nucleus emits an elec- tron. A \(^{210} \mathrm{Bi}\) (bismuth) nucleus at rest undergoes beta decay to \(^{210} \mathrm{Po}\) (polonium). Suppose the emitted electron moves to the right with a momentum of \(5.60 \times 10^{-22} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s} .\) The \(^{210} \mathrm{Po}\) nucleus, with mass \(3.50 \times 10^{-25} \mathrm{kg}\) , recoils to the left at a speed of \(1.14 \times 10^{3} \mathrm{m} / \mathrm{s}\) . Momentum conservation requires that a second particle, called an antineutrino, must also be emitted. Calculate the magnitude and direction of the momentum of the antineutrino that is emitted in this decay.

Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan's weight is 800 \(\mathrm{N}\) , Jane's weight is \(600 \mathrm{N},\) and that of the sleigh is 1000 \(\mathrm{N}\) . They see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity of 5.00 \(\mathrm{m} / \mathrm{s}\) at \(30.0^{\circ}\) above the horizontal (relative to the ice), and Jane jumps to the right at 7.00 \(\mathrm{m} / \mathrm{s}\) at \(36.9^{\circ}\) above the horizontal (relative to the ice). Calculate the sleigh's horizontal velocity (magnitude and direction) after they jump out.

Two friends, Burt and Ernie, are standing at opposite ends of a uniform log that is floating in a lake. The log is 3.0 \(\mathrm{m}\) long and has mass 20.0 \(\mathrm{kg} .\) Burt has mass 30.0 \(\mathrm{kg}\) and Ernie has mass 40.0 \(\mathrm{kg} .\) Initially the log and the two friends are at rest relative to the shore. Burt then offers Ernie a cookie, and Ernie walks to Burt's end of the log to get it. Relative to the shore, what distance has the log moved by the time Ernie reaches Burt? Neglect any horizontal force that the water exerts on the log and assume that neither Burt nor Ernie falls off the log.

A 45.0 -kg woman stands up in a 60.0 -kg canoe 5.00 \(\mathrm{m}\) long. She walks from a point 1.00 \(\mathrm{m}\) from one end to a point 1.00 \(\mathrm{m}\) from the other end (Fig. P8.106). If you ignore resistance to motion of the canoe in the water, how far does the canoe move during this process?

You are standing on a concrete slab that in turn is resting on a frozen lake. Assume there is no friction between the slab and the ice. The slab has a weight five times your weight. If you begin walking forward at 2.00 \(\mathrm{m} / \mathrm{s}\) relative to the ice, with what speed, relative to the ice, does the slab move?

A \(20.0-\mathrm{kg}\) projectile is fired at an angle of \(60.0^{\circ}\) above the horizontal with a speed of 80.0 \(\mathrm{m} / \mathrm{s} .\) At the highest point of its trajectory, the projectile explodes into two fragments with equal mass, one of which falls vertically with zero initial speed. You can ignore air resistance. (a) How far from the point of firing does the other fragment strike if the terrain is level? (b) How much energy is released during the explosion?

A fireworks rocket is fired vertically upward. At its maximum height of \(80.0 \mathrm{m},\) it explodes and breaks into two pieces: one with mass 1.40 \(\mathrm{kg}\) and the other with mass 0.28 \(\mathrm{kg} .\) In the explosion, 860 \(\mathrm{J}\) of chemical energy is converted to kinetic energy of the two fragments. (a) What is the speed of each fragment just after the explosion? (b) It is observed that the two fragments hit the ground at the same time. What is the distance between the points on the ground where they land? Assume that the ground is level and air resistance can be ignored.

A \(12.0-\mathrm{kg}\) shell is launched at an angle of \(55.0^{\circ}\) above the horizontal with an initial speed of 150 \(\mathrm{m} / \mathrm{s} .\) When it is at its highest point, the shell explodes into two fragments, one three times heavier than the other. The two fragments reach the ground at the same time. Assume that air resistance can be ignored. If the heavier fragment lands back at the same point from which the shell was launched, where will the lighter fragment land, and how much energy was released in the explosion?

A Multistage Rocket. Suppose the first stage of a two-stage rocket has total mass \(12,000 \mathrm{kg},\) of which 9000 \(\mathrm{kg}\) is fuel. The total mass of the second stage is \(1000 \mathrm{kg},\) of which 700 \(\mathrm{kg}\) is fuel. Assume that the relative speed \(v_{\text { cx }}\) of ejected material is constant, and ignore any effect of gravity. The effect of gravity is small during the firing period if the rate of fuel consumption is large.) (a) Suppose the entire fuel supply carried by the two-stage rocket is utilized in a single-stage rocket with the same total mass of \(13,000 \mathrm{kg} .\) In terms of \(v_{\mathrm{cx}},\) what is the speed of the rocket, starting from rest, when its fuel is exhausted? (b) For the two-stage rocket, what is the speed when the fuel of the first stage is exhausted if the first stage carries the second stage with it to this point? This speed then becomes the initial speed of the second stage. At this point, the second stage separates from the first stage. (c) What is the final speed of the second stage? (d) What value of \(v_{\mathrm{ex}}\) is required to give the second stage of the rocket a speed of 7.00 \(\mathrm{km} / \mathrm{s} ?\)

A Variable-Mass Raindrop. In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is $$F_{\mathrm{ext}}=\frac{d p}{d t}=m \frac{d v}{d t}+v \frac{d m}{d t}$$ Suppose the mass of the raindrop depends on the distance \(x\) that it has fallen. Then \(m=k x,\) where \(k\) is a constant, and \(d m / d t=k v\) . This gives, since \(F_{\text { ext }}=m g\) , $$m g=m \frac{d v}{d t}+v(k v)$$ Or, dividing by \(k\) $$x g=x \frac{d v}{d t}+v^{2}$$ This is a differential equation that has a solution of the form \(v=\) at, where \(a\) is the acceleration and is constant. Take the initial velocity of the raindrop to be zero. (a) Using the proposed solution for \(v\) , find the acceleration \(a\) . (b) Find the distance the raindrop has fallen in \(t=3.00\) s. (c) Given that \(k=2.00 \mathrm{g} / \mathrm{m},\) find the mass of the raindrop at \(t=3.00 \mathrm{s} .\) (For many more intriguing aspects of this problem, see \(\mathrm{K} .\) S. Krane, American Journal of Physics, Vol. 49 \((1981),\) pp. \(113-117.)\)

In Section 8.5 we calculated the center of mass by considering objects composed of a finite number of point masses or objects that, by symmetry, could be represented by a finite number of point masses. For a solid object whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums of Eqs. (8.28) must be generalized to integrals $$x_{\mathrm{cm}}=\frac{1}{M} \int x d m \quad y_{\mathrm{cm}}=\frac{1}{M} \int y d m$$ where \(x\) and \(y\) are the coordinates of the small piece of the object that has mass \(d m .\) The integration is over the whole of the object. Consider a thin rod of length \(L,\) mass \(M,\) and cross-sectional area \(A .\)Let the origin of the coordinates be at the left end of the rod and the positive \(x\) -axis lie along the rod. (a) If the density \(\rho=M / V\) of the object is uniform, perform the integration described above to show that the \(x\) -coordinate of the center of mass of the rod is at its geometrical center. (b) If the density of the object varies linearly with \(x-\) that is, \(\rho=\alpha x,\) where \(\alpha\) is a positive constant \(-\) calculate the \(x\) -coordinate of the rod's center of mass.