8.Elastic Inelastic Collisions & Linear Momentum

Another reason the center of mass is of such importance is that its motion often displays a remarkable simplicity when compared with the motion of other parts of a system. (page 280)

...

If the objects in a system are not in a line, but are distributed in two dimensions, the center of mass will have both an x coordinate and a y coordinate.

...

A collision is a situation in which two objects ______ one another, and in which the net external force is either zero or negligibly small.

A collision is a situation in which two objects strike one another, and in which the net external force is either zero or negligibly small. (page 267)

A constant linear momentum p is the momentum of an object of mass m that is moving in a ________ line with a velocity v.

A constant linear momentum p is the momentum of an object of mass m that is moving in a straight line with a velocity v. (page 255)

For a completely inelastic collision in one dimension, how would we determine the final velocity?

For a completely inelastic collision in one dimension, how would we determine the final velocity? Because of conservation of momenta, we would equate the initial and final momenta and solve for v₍f₎: v₍f₎ = (m₁v₁,ᵢ + m₂v₂,ᵢ) / (m₁ + m₂) (page 268)

For a completely inelastic collision of two objects of equal mass, with one of the objects initially at rest, how would we determine the final kinetic energy?

For a completely inelastic collision of two objects of equal mass, with one of the objects initially at rest, how would we determine the final kinetic energy? K₍f₎ = ½ (2m) (v / 2 )² = ½ (½mv²) = ½Kᵢ The mass doubles and the speed is halved. The kinetic energy is halved. (page 269)

Internal forces, like all forces, ______ occur in action-reaction pairs.

Internal forces, like all forces, always occur in action-reaction pairs. (page 263)

What is linear momentum?

Linear momentum is the product of the mass m and velocity v of an object: p = mv (Note: vector notation can't be used in Quizlet.)

Is momentum a vector?

Momentum is a vector, with both a magnitude and a direction. (page 255)

∑F = ∆p / ∆t = m (∆v / ∆t) = ma if mass is ________.

∑F = ∆p / ∆t = m (∆v / ∆t) = ma if mass is constant. (page 258)

"In the subatomic world, on the other hand, _______ collisions are common."

"In the subatomic world, on the other hand, elastic collisions are common. Elastic collisions, then, are not merely an ideal that is approached but never attained--they are constantly taking place in nature." (page 273)

"Most collisions in everyday life are rather poor approximations to being elastic --usually there is a significant amount of ______ converted to other forms."

"Most collisions in everyday life are rather poor approximations to being elastic --usually there is a significant amount of energy converted to other forms. However, the collision of objects that bounce off one another with little deformation --like billiard balls, for example--provides a reasonably good approximation to an elastic collision." (page 273)

Because the forces in action-reaction pairs are equal and ________ --due to Newton's third law --internal forces must always sum to zero: ∑F₍int₎ = 0

Because the forces in action-reaction pairs are equal and opposite --due to Newton's third law --internal forces must always sum to zero: ∑F₍int₎ = 0 (page 263)

Because the internal forces cancel, the change in the net momentum is directly related to the net ________ force: ∆p₍net₎ = (∑F₍ext₎) ∆t

Because the internal forces cancel, the change in the net momentum is directly related to the net external force: ∆p₍net₎ = (∑F₍ext₎) ∆t (page 264)

Conservation of momentum for a system of objects: -________ forces have absolutely no effect on the net momentum of a system. -If the net external force acting on a system is zero, its net momentum is conserved.

Conservation of momentum for a system of objects: -Internal forces have absolutely no effect on the net momentum of a system. -If the net external force acting on a system is zero, its net momentum is conserved. (page 264)

Conservation of momentum applies to the net momentum of a ______, not to the momentum of each individual object.

Conservation of momentum applies to the net momentum of a system, not to the momentum of each individual object. All we can say is that the sum of p₁ and p₂ does not change. (page 264)

For collisions in two dimensions, we must _______ the momentum component by component.

For collisions in two dimensions, we must conserve the momentum component by component. We set up a coordinate system and resolve the initial momentum in x and y components. Next, we demand that the final momentum have precisely the same x and y components as the initial momentum. (page 271)

F₍net₎ = ∑F = ∑F₍ext₎ + ∑F₍int₎ Internal and external forces play very _________ roles in terms of how they affect the momentum of a system.

F₍net₎ = ∑F = ∑F₍ext₎ + ∑F₍int₎ Internal and external forces play very different roles in terms of how they affect the momentum of a system. (page 263)

If K₍f₎ = Kᵢ, the collision is said to be _______.

If K₍f₎ = Kᵢ, the collision is said to be elastic. (page 267)

If K₍f₎ ≠ Kᵢ, the collision is said to be _______.

If K₍f₎ ≠ Kᵢ, the collision is said to be inelastic. (page 267)

If an object comes to rest, its final momentum is ____.

If an object comes to rest, its final momentum is zero. (page 256)

If an object falls to the floor and bounces off, its change in momentum is ___.

If an object falls to the floor and bounces off, its change in momentum is 2mv. This is because the momentum in the y direction must first be increased from -mv to 0, then increases again from 0 to mv. (page 256)

If no net forces are acting on an object, final momentum equals initial momentum: p₍f₎ = pᵢ and therefore, its momentum is _________.

If no net forces are acting on an object, final momentum equals initial momentum: p₍f₎ = pᵢ and therefore, its momentum is conserved. (page 262)

If one of two objects connected to the ends of a rod has more mass than the other, the center of mass is closer to the _______ object.

If one of two objects connected to the ends of a rod has more mass than the other, the center of mass is closer to the heavier object. (page 278)

If the initial momentum and the final momentum sum to zero, does that mean that the kinetic energy sums to zero?

If the initial momentum and the final momentum sum to zero, it doesn't mean that the kinetic energy sums to zero. Kinetic energy is never negative. (page 266)

If the net force acting on an object is zero, its change in momentum is also ____.

If the net force acting on an object is zero, its change in momentum is also zero. (page 262)

If two objects connected to the ends of a rod have the same mass, the center of mass is at the ________ of the rod, since this is where it balances.

If two objects connected to the ends of a rod have the same mass, the center of mass is at the midpoint of the rod, since this is where it balances. (page 278)

What is impulse?

Impulse is average force times the length of time, ∆t, that two objects are in contact: I = Fₐᵥ ∆t (page 258)

In a collision between two objects of different mass, a significant amount of momentum can be transferred from the _____ object to the _____ object.

In a collision between two objects of different mass, a significant amount of momentum can be transferred from the large object to the small object. Even though the total momentum is conserved, the small object can be given a speed that is significantly larger than any of the initial speeds. (page 275)

In a two-dimensional elastic collision, if we are given the final speed and direction of one of the objects, we can find the speed and direction of the other object using ______ conservation and ________ conservation.

In a two-dimensional elastic collision, if we are given the final speed and direction of one of the objects, we can find the speed and direction of the other object using energy conservation and momentum conservation. (page 276)

In an elastic collision in one dimension where mass m₁ is moving with an initial velocity v₀, and mass m₂ is initially at rest, the velocities of the masses after the collision are...?

In an elastic collision in one dimension where mass m₁ is moving with an initial velocity v₀, and mass m₂ is initially at rest, the velocities of the masses after the collision are: v₁₍f₎ = (m₁ − m₂ / m₁ + m₂) v₀ v₂₍f₎ = (2m₁ / m₁ + m₂) v₀ (page 273)

In an elastic collision in one dimension where mass m₁ is moving with an initial velocity v₀, and mass m₂ is initially at rest: -the final velocity of m₁ can be positive, negative, or zero, depending on whether m₁ is greater than, ____ than, or equal to m₂, respectively. The final velocity of m₂, however, is always positive.

In an elastic collision in one dimension where mass m₁ is moving with an initial velocity v₀, and mass m₂ is initially at rest: -the final velocity of m₁ can be positive, negative, or zero, depending on whether m₁ is greater than, less than, or equal to m₂, respectively. The final velocity of m₂, however, is always positive. (page 273)

In an elastic collision, momentum and _______ energy are conserved.

In an elastic collision, momentum and kinetic energy are conserved. That is: p₍f₎ = pᵢ and K₍f₎ = Kᵢ (page 273)

How are collisions categorized?

In general, collisions are categorized according to what happens to the kinetic energy of the system. (page 267)

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very large and the mass of the other object at rest is much smaller and approaches zero, we substitute ____ for m₂ into the equations for the final velocities.

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very large and the mass of the other object at rest is much smaller and approaches zero, we substitute zero for m₂ into the equations for the final velocities. (Please see p. 275 for correct notation). v₁₍f₎ = (m₁ / m₁) v₀ = v₀ v₂₍f₎ = (2m₁ / m₁) v₀ = 2v₀

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very small and approaches zero, but the mass of the other object at rest is much larger, the lighter object is reflected, heading ________ at its initial speed.

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very small and approaches zero, but the mass of the other object at rest is much larger, the lighter object is reflected, heading backward at its initial speed. (page 275)

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very small and approaches zero, but the mass of the other object at rest is much larger, we substitute ____ for m₁ into the equations for the final velocities.

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very small and approaches zero, but the mass of the other object at rest is much larger, we substitute zero for m₁ into the equations for the final velocities (page 274): v₁₍f₎ = (0 − m₂ / 0 + m₂) v₀ = (-m₂ / m₂ ) v₀ = −v₀ v₂₍f₎ = [(2⋅0) / (0 + m₂)] v₀ = 0

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very large and the mass of the other object at rest is much smaller and approaches zero, the speed of the larger object is unaffected, but the smaller object rebounds at _____ the speed of the larger object.

In the special case of an elastic collision in one dimension where the mass of the object moving with v₀ is very large and the mass of the other object at rest is much smaller and approaches zero, the speed of the larger object is unaffected, but the smaller object rebounds at twice the speed of the larger object. (page 275)

In the special case of an elastic collision in one dimension where the masses of the two objects are equal, m₁ has v₀ and m₂ is initially at rest: -the final velocity of m₁ will be ____. -the final velocity of m₂ will be v₀.

In the special case of an elastic collision in one dimension where the masses of the two objects are equal, m₁ has v₀ and m₂ is initially at rest: -the final velocity of m₁ will be zero. -the final velocity of m₂ will be v₀. (The two objects have exchanged velocities.) (page 274)

In the special case where objects stick together after the collision, we say that the collision is __________ inelastic.

In the special case where objects stick together after the collision, we say that the collision is completely inelastic. (page 268)

How do we express the application of Newton's Second Law to a system of particles?

Newton's Second Law for a system of particles: MA₍cm₎ = F₍net,ext₎ Total mass of the system times acceleration of the center of mass equals the net external force. (page 281)

One of the reasons the center of mass is so special is the fact that, in many ways, a system behaves as if ___ of its mass were concentrated there.

One of the reasons the center of mass is so special is the fact that, in many ways, a system behaves as if all of its mass were concentrated there.

Since momentum is a vector, the total momentum of a system of objects is the ______ sum of the momenta of all the objects.

Since momentum is a vector, the total momentum of a system of objects is the vector sum of the momenta of all the objects. (page 256)

What is the Momentum-Impulse Theorem?

The Momentum-Impulse Theorem states that in general impulse is just the change in momentum: I = Fₐᵥ ∆t = ∆p (page 259)

What is the SI unit for momentum?

The SI unit for linear momentum is: kg ⋅ m/s or mass times the units of velocity. There is no special name for the unit of momentum. (page 255)

What is the SI unit of an impulse?

The SI unit of an impulse is newtons times seconds, or N ⋅s or kg ⋅m/s This is the same as the unit of momentum. (page 259)

The center of mass is at the geometric center of a uniform object, even if there is no ____ at that location.

The center of mass is at the geometric center of a uniform object, even if there is no mass at that location. (page 279)

The center of mass of a system accelerates precisely as if it were a point ________ of mass M acted on by the force F₍net,ext₎.

The center of mass of a system accelerates precisely as if it were a point particle of mass M acted on by the force F₍net,ext₎. For this reason, the motion of the center of mass can be quite simple compared to the motion of its constituent parts. (page 283)

The center of mass of a system of masses is the _____ where the system can be balanced in a uniform gravitational field.

The center of mass of a system of masses is the point where the system can be balanced in a uniform gravitational field. (page 278)

The change in momentum during a time interval ∆t is...?

The change in momentum during a time interval ∆t is: ∆p = (∑F) ∆t (page 262)

The fact that the internal forces always cancel means that the net force acting on a system of objects is simply the sum of the ________ forces acting on it: F₍net₎ = ∑F₍ext₎ + ∑F₍int₎ = ∑F₍ext₎

The fact that the internal forces always cancel means that the net force acting on a system of objects is simply the sum of the external forces acting on it: F₍net₎ = ∑F₍ext₎ + ∑F₍int₎ = ∑F₍ext₎ (page 263)

The fact that the momentum of a system is _________ during a collision does not necessarily mean that the system's kinetic energy is conserved.

The fact that the momentum of a system is conserved during a collision does not necessarily mean that the system's kinetic energy is conserved. (page 267)

What is the formula for the center of mass of two objects?

The formula for the center of mass of two objects is: X₍cm₎ = (m₁x₁ + m₂x₂) / m₁ + m₂ = (m₁x₁ + m₂x₂) / M where M is the total mass. This is the weighted average of two positions. (page 278)

What is the formula for the x coordinate of the center of mass?

The formula for the x coordinate of the center of mass is: X₍cm₎ = (m₁x₁ + m₂x₂ + ⋅⋅⋅) / m₁ + m₂ + ⋅⋅⋅ = ∑mx / M A similar formula would be used for the y coordinate. (page 279)

What are the initial and final momenta of two objects in a completely inelastic collision, where the two objects stick together?

The initial and final momenta of two objects in a completely inelastic collision, where the two objects stick together, are: pᵢ = m₁v₁,ᵢ + m₂v₂,ᵢ p₍f₎ = (m₁ + m₂)v₍f₎

What is the more general form of Newton's second law, which holds even if the mass changes?

The more general form of Newton's second law, which holds even if the mass changes, is expressed in terms of momentum: ∑F = ∆p / ∆t or, the net force acting on an object is equal to the change in its momentum divided by the time interval during which the change occurs. (page 258)

The movement in the opposite direction that occurs because of Newton's _____ law is called recoil.

The movement in the opposite direction that occurs because of Newton's third law is called recoil. (page 265)

The ___ force acting on a system of objects is the sum of forces applied from outside the system (external forces) and forces acting between objects within the system (internal forces).

The net force acting on a system of objects is the sum of forces applied from outside the system (external forces) and forces acting between objects within the system (internal forces). (page 263)

The speed of separation after a head-on elastic collision is always _____ __ the speed of approach before the collision.

The speed of separation after a head-on elastic collision is always equal to the speed of approach before the collision. (page 275)

The total mass of a system, M, times the acceleration of the center of mass, A₍cm₎, is simply the total _____ acting on the system.

The total mass of a system, M, times the acceleration of the center of mass, A₍cm₎, is simply the total force acting on the system. The total force acting on a system is the same as the net external force, since the internal forces cancel. (page 281)

The total momentum of the universe is _________.

The total momentum of the universe is conserved. (page 266)

What is the unit of thrust?

The unit of thrust is the newton, N. (page 284)

What is thrust?

Thrust is force exerted by change in mass over change in time multiplied by velocity: thrust = (∆m / ∆t) v (page 284)

How do we find the acceleration of the center of mass?

To find the acceleration of the center of mass, multiply the mass of each object in a system by its acceleration. Add all the products together and divide by the total mass. A₍cm₎ = (m₁a₁ + m₂a₂ + ⋅⋅⋅) / (m₁ + m₂ + ⋅⋅⋅) = ∑ma / M (page 281)

How do we find the velocity of the center of mass?

To find the velocity of the center of mass, multiply the mass of each object in a system by its velocity. Add all the products together and divide by the total mass. V₍cm₎ = (m₁v₁ + m₂v₂ + ⋅⋅⋅) / (m₁ + m₂ + ⋅⋅⋅) = ∑mv / M (page 280)