Chp. 7 - Momentum

Cars used to be built as rigid as possible to withstand collisions. Today, though, cars are designed to have "crumple zones" that collapse upon impact. What is the advantage of this new design?

"Crumple zones" are similar to airbags. As the crumple zone collapses, the process increases the time over which a stopping force acts on the car. This allows for a smaller average force to act on the car, which reduces the chance for a serious injury to the occupants. Increased stopping time lowers average force and can help reduce injury. Modern vehicles are designed to increase stopping time in a number of ways, including crumple zones and airbags.

Newton's Second Law

(In terms of momentum) the net external force equals the change in momentum of a system divided by the time over which it changes. Fnet = ∆p/∆t Ex: A tennis ball (mass = 0.1 kg) is thrown and strikes the floor with a velocity of 5 m/s downwards and bounces back at a final velocity of 3m/s upwards. As the ball approaches the floor it has an initial momentum p→i. When it moves away from the floor it has a final momentum p→f. The bounce on the floor can be thought of as a collision taking place where the floor exerts a force on the tennis ball to change its momentum.

Center of Mass

(of an extended object or group of objects) is the point at which the net force can be considered to act, for purposes of determining the translational motion of the object as a whole. Ex: the center of mass of a uniform disc shape would be at its center. Or the CM of a person standing still is by their belly button.

The space shuttle, in circular orbit around the Earth, collides with a small asteroid which ends up in the shuttle's storage bay. For this collision, (a) only momentum is conserved. (b) only kinetic energy is conserved. (c) both momentum and kinetic energy are conserved. (d) neither momentum nor kinetic energy is conserved.

A The asteroid and shuttle collided and "stuck together," and have the same final speed. This is a textbook example (ha ha) of a completely inelastic collision. During the collision, momentum is conserved, but kinetic energy is not.

An astronaut is a short distance away from her space station without a tether rope. She has a large wrench. What should she do with the wrench to move toward the space station? (a) Throw it directly away from the space station. (b) Throw it directly toward the space station. (c) Throw it toward the station without letting go of it. (d) Throw it parallel to the direction of the station's orbit. (e) Throw it opposite to the direction of the station's orbit.

A The momentum of the astronaut-wrench system is initially zero.If she throws the wrench but holds onto it, she'll momentarily jerk backwards, but come to rest again just as the wrench does.If the astronaut throws a wrench directly away from the station, she will move in the opposite direction, toward the station, with an equal and opposite momentum as the wrench. This conserves the momentum of the person-wrench system, which was initially zero.

You are lying in bed and want to shut your bedroom door. You have a bouncy "superball" and a blob of clay, both with the same mass. Which one would be more effective to throw at your door to close it? (a) The superball. (b) The blob of clay. (c) Both the same. (d) Neither will work.

A The moral is not to ignore the vector nature of momentum and impulse.The ball and clay each have the same momentum just before they contact the door.The change in momentum of the ball is greater when it rebounds, compared to the clay's change in momentum which is simply brought to a dead stop. The ball was not only brought to rest, but the greater impulse delivered to it reversed its momentum and sent it back the way it came.Each object delivers an impulse to the door that is equal and opposite to the impulse it received from the door. The superball delivers a greater impulse to the door, making it more effective for the task.

A golf ball and an equal-mass bean bag are dropped from the same height and hit the ground. The bean bag stays on the ground while the golf ball rebounds. Which experiences the greater impulse from the ground? (a) The golf ball. (b) The bean bag. (c) Both the same. (d) Not enough information.

A The moral is not to ignore the vector nature of momentum and impulse.The beanbag and golf ball each have the same momentum just before they contact the ground.The change in momentum of the golf ball is greater when it rebounds, compared to the beanbag's change in momentum, which is simply brought to a dead stop. The golf ball was not only brought to rest, but the greater impulse reversed its momentum and sent it back the way it came.

Inelastic Collision

A collision where KE is not conserved. Macroscopic collisions are generally inelastic. Completely inelastic collisions is one where objects stick together. KE1 + KE2 = KE1' + KE2' + Elost If problem is inelastic the problem usually says that the objects stick together. In this case, some of the KE is transferred to other forms of internal energy so there is not conservation of KE.

Why is the CM of a 1-m length of pipe at its midpoint, whereas this is not true for your arm or leg?

A pipe is uniform throughout, while a limb is not. A 1-meter length of pipe has uniform mass density. By symmetry, its CM is at the geometric center, the midpoint.Your arm and leg aren't uniform; the cross-section differs, and the density of muscle, fat and bone isn't the same.Generally, the CM of your arm or leg is closer to the torso than the midpoint of the limb.

Isolated System

A system where the only (significant) forces are those between the objects in the system. Ex: The collision of two balls on the billiards table. The collision occurs in an isolated system as long as friction is small enough that its influence upon the momentum of the billiard balls can be neglected. If so, then the only unbalanced forces acting upon the two balls are the contact forces that they apply to one another. These two forces are considered internal forces since they result from a source within the system - that source being the contact of the two balls. For such a collision, total system momentum is conserved.

Some people think that momentum is just inertia. A) Compare the inertia of a Mack Truck to Jennifer on Rollerblades: Is this always true?Explain. B) Compare the momentum of a Mack Truck to Jennifer on Rollerblades: Is this always true?Explain:

A) The Mack Truck always has more inertia (b/c Mack Truck has a larger mass). B) Mack Truck would always have more inertia if it was moving, but if it was at rest, Jennifer would be going super fast.

In one type of nuclear radioactive decay, an electron and a recoil nucleus are emitted but often do not separate along the same line. Use conservation of momentum in two dimensions to explain why this implies the emission of at least one other particle (it came to be called a "neutrino").

Assume the nucleus is initially stationary, and has zero momentum. If the nucleus broke into only two particles, then by conservation of momentum, the momenta of the two decay products would have to cancel, to keep the final momentum equal to zero. For these 2 momentum vectors to add to zero, they must lie along a line. Observing that the recoil nucleus and the electron don't fly out in exactly opposite directions implies that at least one other particle must carry some momentum.

Why are not collisions all instantaneous?

At the point of impact, objects deform; no matter how rigid they may seem.

. A small boat coasts at constant speed under a bridge. A heavy sack of sand is dropped from the bridge onto the boat. The speed of the boat (a) increases. (b) decreases. (c) does not change. (d) Without knowing the mass of the boat and the sand, we can't tell.

B Though the sand is dropped onto the boat vertically, it does affect the boat's horizontal motion. Initially the sand's horizontal velocity is zero. It must be accelerated from rest to the final horizontal velocity of the boat. There is a force between the boat deck and the sandbag, pushing the bag forward.By Newton's third law, the boat feels an equal but opposite force from the sand, directed backward, and this causes the boat speed to decrease.

The longer the impact time of an impulse, the smaller the force can be for the same momentum change, and hence the smaller the deformation of the object on which the force acts. On this basis, explain the value of air bags, which are intended to inflate during an automobile collision and reduce the possibility of fracture or death.

By increasing the time of impact, the overall force of the impact on the passenger is lessened. As the driver slowly "sinks into" the airbag, the process increases the time over which a stopping force acts on the driver. This allows for a smaller force to act on the driver, which reduces the chance for a serious injury.

Two identical billiard balls traveling at the same speed have a head-on collision and rebound. If the balls had twice the mass, but maintained the same size and speed, how would the rebound be different? (a) At a higher speed. (b) At slower speed. (c) No difference.

C The system's momentum is zero before the collision because the identical balls are moving at the same speed and have oppositely directed velocities. Small masses, system momentum before the collision is zero. The system's momentum is still zero after the collision by conservation of momentum. The identical balls are still moving at the same speed as each other, with oppositely directed velocities.Small masses, system momentum after the collision is zero. Now let the masses be doubled. A ball's momentum and its kinetic energy are proportional to its mass. Doubling each ball's mass means that each momentum term in the conservation equation is doubled. However, this factor of two can be canceled out in all the equations, bringing us back to the original equations of motion.Large masses, system momentum before the collision is zero. Large masses, system momentum after the collision is zero Doubling the masses doesn't affect the final velocity.

A bowling ball hangs from a 1.0-m-long cord, Fig. 7-30: (i) A 200-gram putty ball moving hits the bowling ball and sticks to it, causing the bowling ball to swing up; (ii) a 200-gram rubber ball moving hits the bowling ball and bounces straight back at nearly causing the bowling ball to swing up. Describe what happens. (a) The bowling ball swings up by the same amount in both (i) and (ii). (b) The ball swings up farther in (i) than in (ii). (c) The ball swings up farther in (ii) than in (i). (d) Not enough information is given; we need the contact time between the rubber ball and the bowling ball.

C The final height of the bowling ball depends upon the impulse delivered to it by each object.The moral is not to ignore the vector nature of momentum and impulse.The rubber ball and putty each have the same momentum just before they contact the bowling ball. The change in momentum of the rubber ball is greater when it rebounds, compared to the putty's change in momentum, which is simply brought to a dead stop. The rubber ball was not only brought to rest, but the greater impulse delivered to it reversed its momentum and sent it back the way it came.Each object delivers an impulse to the bowling ball that is equal and opposite to the impulse it received from the bowling ball. The rubber ball imparts a greater impulse to the bowling ball than the putty does, sending the bowling ball farther up.

A baseball is pitched horizontally toward home plate with a velocity of 110 km h. In which of the following scenarios does the baseball have the largest change in momentum? (a) The catcher catches the ball. (b) The ball is popped straight up at a speed of 110 km h. (c) The baseball is hit straight back to the pitcher at a speed of 110 km h. (d) Scenarios (a) and (b) have the same change in momentum. (e) Scenarios (a), (b), and (c) have the same change in momentum.

C The moral is not to ignore the vector nature of momentum and impulse.Consider the ball's momentum just when it arrives at home plate. Call the magnitude of the momentum p.If the ball is stopped, its final momentum is zero and the change in momentum (impulse) has magnitude p.If the ball is popped straight up at the same speed, its final momentum is straight upward with magnitude p, and the change in momentum (impulse) has magnitude p√2.If the ball is hit backward at the same speed, its final momentum is backward with magnitude p, and the change in momentum (impulse) has magnitude 2p.

A railroad tank car contains milk and rolls at a constant speed along a level track. The milk begins to leak out the bottom. The car then (a) slows down. (b) speeds up. (c) maintains a constant speed. (d) Need more information about the rate of the leak.

C You might think that as the milk drains the tank car's speed increases. However, the crucial difference here is that when the milk drops away, it has the same forward horizontal velocity that it had when it was inside the tank.No horizontal forces act on the spilled milk, so it exerts no horizontal force on the tank car and the remaining milk.By Newton's first law, the car's speed doesn't change.

Impulse

Change in momentum in a collision. ∆p = F∆t

Explain, on the basis of conservation of momentum, how a fish propels itself forward by swishing its tail back and forth.

Consider the fish and water as our system. The swishing tail pushes water backward. The system's momentum is conserved. The moving water gains momentum one way, and the fish gains momentum the other way, i.e., it moves in the opposite direction, or forward.

A truck going has a head-on collision with a small car going Which statement best describes the situation? (a) The truck has the greater change of momentum because it has the greater mass. (b) The car has the greater change of momentum because it has the greater speed. (c) Neither the car nor the truck changes its momentum in the collision because momentum is conserved. (d) They both have the same change in magnitude of momentum because momentum is conserved. (e) None of the above is necessarily true.

D Assume that the truck and car are part of the system. Assuming there are no net external forces, or that they act over a negligibly short time, momentum during the collision is conserved (i.e., the change in system momentum is zero). Any momentum gained by the truck is lost by the car.In other words, their individual momentum changes are equal and opposite.

A small car and a heavy pickup truck are both out of gas. The truck has twice the mass of the car. After you push both the car and the truck for the same amount of time with the same force, what can you say about the momentum and kinetic energy (KE) of the car and the truck? Ignore friction. (a) They have the same momentum and the same KE. (b) The car has more momentum and more KE than the truck. (c) The truck has more momentum and more KE than the car. (d) They have the same momentum, but the car has more kinetic energy than the truck. (e) They have the same kinetic energy, but the truck has more momentum than the car.

D This question is a good review of the relationships among force, time, momentum, work, and KE.Impulse delivered is the force multiplied by the time over which the force is acting. For an object with zero initial momentum (starting at rest), the impulse is the final momentum.The same force acts for the same time on both vehicles, so they have the same final momentum.The KE of an object can be expressed in terms of its momentum. KE=12mv2=(mv)22m=p22m We see that if 2 objects have the same momentum, the less massive object has a larger kinetic energy.Choice D summarizes the situation.

Center of mass is almost the same as center of gravity. How is CM and CG different?

Different only when an object is so large that the gravitational field is not equal on the whole object. Ex: a mountain is so tall that the average pull due to gravity is not the same as where the average mass us.

What are examples of when the center of mass lies outside the body?

Doughnut, boomerang, banana, empty pop can

We claim that momentum is conserved. Yet most moving objects eventually slow down and stop. Explain.

For momentum to be conserved in a system, the system is assumed to be closed, i.e., there are no external forces acting.If we consider, say, a sliding block on a table, then air resistance and friction act on the block. These outside/external forces change the system's (i.e., block's) momentum.

Is there more force on a wall if something hits and sticks, or if something hits and bounces back?

Hits something and BOUNCES back.

The speed of a tennis ball on the return of a serve can be just as fast as the serve, even though the racket isn't swung very fast. How can this be?

If a tennis ball bounces elastically off an unmoving, much larger-mass tennis racquet, it reverses its normal component of velocity and can rebound with practically all of its original speed. If the racquet is moving toward the ball, even just a little, the ball can rebound with a kinetic energy even greater than it came in with.

It is said that in ancient times a rich man with a bag of gold coins was stranded on the surface of a frozen lake. Because the ice was frictionless, he could not push himself to shore and froze to death. What could he have done to save himself had he not been so miserly?

If he had been less miserly, he could have thrown a coin horizontally. He would move in the opposite direction, with an equal and opposite momentum. This conserves the momentum of the person-coin system, which was initially zero. He would be moving much more slowly than the thrown coin, but since the lake is frictionless, he'd eventually reach shore. Or, he could be less cheap and throw more coins to speed things up.

Why can a batter hit a pitched baseball farther than a ball he himself has tossed up in the air?

If the bat were a stationary, large-mass object such as a wall, the rebounded baseball would travel away only about as fast as it came in. A pitched ball with a faster incoming speed would bounce away faster. A similar argument shows why a fast incoming baseball will rebound from a moving bat much faster than a slow incoming baseball.

How can little objects have a lot of momentum?

If they have a really high velocity.

How are rockets propelled (according to linear momentum)?

In the reference frame of the rocket, before any fuel is ejected , the total momentum of the rocket plus fuel is 0. When the fuel burns, the total momentum remains uncharged: the backward momentum of the expelled gases is just balanced by the forward momentum gained by the rocket itself. Thus a rocket can accelerate in space.

Explain using initial and final momentum of the system, why an inflated, untied balloon flies across the room:

Initally, the momentum of the balloon is zero. So if the gases have a backward momentum, then the balloon must have a forward momentum so the total remains zero to conserve momentum. Alternative: The air inside the balloon shoots out of the balloon's opening. The system's momentum is conserved. The escaping gas gains momentum one way, and the balloon and its remaining gas gain momentum the other way, i.e., they move in the opposite direction.

Bob and Jim decide to play tug-of-war on a frictionless (icy) surface. Jim is considerably stronger than Bob, but Bob weighs 160 lb whereas Jim weighs 145 lb. Who loses by crossing over the midline first? Explain.

Jim, the lighter fellow, will lose. Consider Bob, Jim, and the rope as a system. Because there is no net external force on the system and the system is initially stationary, the center of mass (CM) does not move. The CM starts out closer to Bob (because he is more massive), i.e., the CM is on Bob's side of the midline.The CM doesn't move, so as the men pull, they will eventually meet at the center of mass. Since the CM is on Bob's side, Jim will have crossed the midline to get there.This problem may also be analyzed using Newton's third and second laws.

Elastic Collision

KE and momentum conserved - no mechanical energy lost ex: pool balls, ideal gas particles. NO KE lost!

Why do you tend to lean backward when carrying a heavy load in your arms?

Leaning back moves your center of mass, the heavy object, to be above your feet. To keep your balance, your center of mass (CM) should be located above your feet. If you carry a heavy load in your arms, your CM shifts toward the load, moving in front of your body, possibly toppling you forward. You instinctively lean back to keep your CM above your feet.

How does an isolated system relate to linear momentum?

Momentum is only conserved within a system; For instance, if you had two cars in a collision, their momentums total before and after are the same, but if you had a force from outside the system - say a wind or another car - they could change that total because Fxt = ∆mv, their force could add or take away from the total momentum. So when we say momentum is conserved with have to draw a box around the system and not have any outside forces.

Does a train always have a lot of momentum?

No; Could be at rest or moving very slow.

If two different objects have the same momentum, do they necessarily have the same kinetic energy?

No; a product of the mass × velocity is not nessecarily the same as mass × velocity^2. 2*5 = 10 5*2 = 10 1/2(2)(5)^2 = 25 1/2(5)(2)^2 = 10

Linear Momentum

Product of an object's mass and velocity. p = mv unit = kgm/s (has no special name) :(

How can you reduce the force felt by reducing the impulse?

Reducing ∆p(mvf - mvi); you could have lower Vi or less mass

Explain with vector math and their formulas, why momentum is a vector but kinetic energy is not.

Since velocity is a vector quantity, mass × velocity would be a vector and has direction ( + or - ). KE is not a vector since the velocity is squared, it loses its direction and can only be positive.

What are examples of objects with high momentum?

Speeding freight train (although momentum if the train stopped = 0) or a high speed bullet.

A boy stands on the back of a rowboat and dives into the water. What happens to the boat as he leaves it? Explain.

The boat moves in the opposite direction, i.e., forward, when the boy dives off the back. This conserves the momentum of the boy-boat system, which was initially zero.

How can a rocket change direction when it is far out in space and essentially in a vacuum?

The engine can expel hot exhaust, compressed gas, or ions in a given direction. The gas-rocket system is isolated and its momentum stays constant. If some mass is expelled, say, to the left, then the rocket and the remaining gas moves to the right to keep the overall momentum the same (this situation can also be analyzed using Newton's third law).

If only an external force can change the momentum of the center of mass of an object, how can the internal force of the engine accelerate a car?

The external force from the road accelerates the car. The engine spins the wheels, which push backward on the road surface. Newton's third law predicts that the road surface pushes forward on the wheels, which accelerates the car.

A person stands under an umbrella during a rain shower. A few minutes later the raindrops turn to hail though the number of "drops" hitting the umbrella per time and their speed and mass remains the same. Is the force required to hold the umbrella in the hail the same as, more than or less than the force required in the rain? Explain!

The force is great w/ hail since the change in momentum is greater: the hail bounces while the rain just splats and stops.

What happens in a collision w/ a given initial and final velocity?

The impulse does not change for two different scenarios, but the force felt does.

Conservation of KE

The kinetic energy of interacting bodies or particles in a closed system remains constant. KEi+PEi=KEf+PEf

A satellite explodes in outer space far from any other body, sending thousands of pieces in all directions. How does the linear momentum of the satellite before the explosion compare with the total linear momentum of all the pieces after the explosion?

The momentum before the explosion is zero. So after the explosion the total momentum of all the pieces cancel, so it is also zero.

When a person jumps from a tree to the ground, what happens to the momentum of the person upon striking the ground?

The person sticks to the Earth, and the Earth picks up some momentum. The total momentum (person + Earth) is unchanged. Because of the Earth's huge mass, its extra momentum is not noticeable.

A rocket following a parabolic path through the air suddenly explodes into many pieces. What can you say about the motion of this system of pieces?

The pieces of the rocket fly apart willy-nilly, but the CM of these moving pieces will follow the same path that the rocket would have originally followed. The rocket is an isolated system, and its center of mass (CM) would follow a parabolic path under the influence of Earth's gravity (an external force).The explosion consists of forces that are internal to the system, so these cannot affect the motion of the CM.

A hover fly is happily maintaining a fixed position about 10 ft above the ground when an elephant charges out of the bush and collides with it. The fly bounces elastically off the forehead of the elephant. If the initial speed of the elephant is vo, what is the speed of the fly after the collision equal to?

The speed of the fly is 2x the initial speed of the elephant, because the flies mass is negligible.

If momentum changes, what variable do is changed?

Velocity.

At a hydroelectric power plant, water is directed at high speed against turbine blades on an axle that turns an electric generator. For maximum power generation, should the turbine blades be designed so that the water is brought to a dead stop, or so that the water rebounds?

The turbine blades should cause the water to rebound. The change in momentum of the water is greater when it rebounds than if the water is simply brought to a dead stop. The water is not only brought to rest, but the force is great enough to reverse its momentum and send it backwards.Because the water has a greater change in its momentum, the turbine blades have a greater change in their momentum, i.e., they spin faster.

If a 20-passenger plane is not full, sometimes passengers are told they must sit in certain seats and may not move to empty seats. Why might this be?

The weight of the passengers must be balanced to keep the plane's center of gravity in the correct place. For a small plane, where the passengers make up a significant fraction of the total mass, they may be told to sit in certain seats to keep the plane balanced.

Two children float motionlessly in a space station. The 20-kg girl pushes on the 40-kg boy and he sails away at The girl (a) remains motionless; (b) moves in the same direction at 1.0 m/s (c) moves in the opposite direction at 1.0 m/s (d) moves in the opposite direction at 2.0 m/s (e) none of these.

There are no external forces acting on the pair, so their momentum is conserved. The initial momentum is zero.Let's say the boy moves to the right. His final momentum is mv = (40 kg)(1 m/s).The girl's final momentum must be (20 kg)(-2 m/s) so that their net final momentum is still zero.In other words, the girl moves in the opposite direction at a speed of 2.0 m/s.

True or false: The more something deforms, the less average force?

True.

Describe a collision in which all kinetic energy is lost.

Two equal-mass clay blobs that initially have equal and opposite momenta, colliding head-on, will come to a dead stop and all of the initial kinetic energy is lost. The initial momentum of the system is zero, so the final momentum is zero. The blobs stick together, so the final momentum can be zero only if the combined blob is stationary.

Conservation of Momentum

When there are NO EXTERNAL FORCES, momentum of a system is conserved. - Usually used in collisions. Formula: m1Vi + m2vi = m1Vf + m2Vf

If a falling ball were to make a perfectly elastic collision with the floor, would it rebound to its original height? Explain.

Yes. By definition, in a perfectly elastic collision, KE is conserved. If we ignore the incredibly small kinetic energy given to the Earth by the collision, the ball has the same KE just after the bounce as it had just before the bounce. This KE is transformed back into gravitational potential energy, so it will rebound to the same original height.

Is it possible for an object to receive a larger impulse from a small force than from a large force? Explain.

Yes. Impulse is the force multiplied by the time over which it is acting, i.e., the area under the force-time curve.A small force acting for a long time can deliver a huge impulse, compared to a large force acting for a brief time.

A very elastic "superball" is dropped from a height h onto a hard steel plate (fixed to the Earth), from which it rebounds at very nearly its original speed. (a) Is the momentum of the ball conserved during any part of this process? (b) If we consider the ball and the Earth as our system, during what parts of the process is momentum conserved? (c) Answer part (b) for a piece of putty that falls and sticks to the steel plate.

a. No, the momentum of the ball alone is not conserved during any part of the process. An external force, gravity, acts on the ball at all times. Also, during the collision there is a strong, sharp upward force acting on the ball. So if the ball is the system, there are always external forces, so its momentum is never conserved. b. Yes, now that we include the Earth in the system, all of the forces are internal. The momentum of the Earth-ball system is conserved during the entire process. c. The answer is unchanged. If the system is the blob of putty and the Earth, the total momentum is conserved during the entire process.

A squash ball hits a wall at a 45° angle as shown in Fig. 7-29. What is the direction (a) of the change in momentum of the ball, (b) of the force on the wall?

a. The change in momentum of the ball is directed perpendicular to the wall, directly to the left in the diagram. This can be proven by subtracting the initial momentum vector from the final momentum vector b. The force on the ball is to the left, in the same direction as the impulse. By Newton's third law, the force on the wall is to the right.