Momentum

Introduction to Momentum

Momentum can be defined as "mass in motion." All objects have mass; so if an object is moving, then it has momentum - it has its mass in motion.

The amount of momentum which an object has is dependent upon two variables: how much stuff is moving and how fast the stuff is moving. Momentum depends upon the variables mass and velocity.

Definition

• The momentum of an object is equal to the mass of the object times the velocity of the object.

Momentum = mass • velocityp = m • v

• The units for momentum would be mass units times velocity units. The standard unit of momentum is the kg•m/s.

• Momentum is a vector quantity.

Which has more momentum, an airplane taxiing down the runway at 50 mph or a SUV traveling at 50 mph?

The airplane because it has a greater mass.

If the airplane comes to rest and the SUV continues moving at 50 mph, which has more momentum?

The SUV because it is moving. The momentum of the airplane at rest is 0.

ImpulseAny object with momentum is going to be

hard to stop. • To stop an object in motion, it is

necessary to apply a force against its motion for a given period of time.

As the force acts upon the object for a given amount of time, the object's velocity is changed; and hence, the object's momentum is changed.

If the force acts opposite the object's motion, it slows the object down. If a force acts in the same direction as the object's motion, then the force speeds the object up.

Either way, a force will change the velocity of an object. And if the velocity of the object is changed, then the momentum of the object is changed.

• The quantity Force • time is known as impulse.

When a force acts on an object for some amount of time (an impulse exists), the object’s momentum will change.

In racket and bat sports, hitters are often encouraged to follow-through when striking a ball. High speed films of the collisions between bats/rackets and balls have shown that the act of following through serves to increase the time over which a collision occurs.

This increase in time does not increase the force. The force in hitting is dependent upon how hard the hitter swings the bat or racket, not the time of impact.

The follow-through increases the time of collision and subsequently contributes to an increase in the velocity change of the ball. By following through, a hitter can hit the ball in such a way that it leaves the bat or racket with more velocity (i.e., the ball is moving faster).

Impulse-Momentum Theorem

Impulse = Change in momentum

Say you were traveling down the road at 30 mph and you were involved in a wreck. You will experience a change in your momentum.

There is nothing auto-makers can do to make you not get into accidents. If you are involved in a wreck, you will experience a change in momentum.

The air bag of a car cannot affect the change in your momentum, nor can they affect your impulse however, air bags allow you to come to rest over a longer period of time. This means the force on you will be less.

Let’s look at some numbers. Say a 2 kg object traveling at 10 m/s comes to rest. The change in momentum of the object is

mv = mvf - mvi

mv = (2 kg)(0 m/s) – (2 kg)(10 m/s)

mv = -20 kgm/s

The impulse momentum theorem says that the impulse on the object is equal to the change in momentum.

Ft = mv

Ft = -20 kgm/s

We don’t know what the force and time are for the object. We could have:Force = 20 N and time = 1sForce = 5 N and time = 4 sForce = 2 N and time = 10 s.

We just don’t know but we do know the force and time must have a product of 20.

Air bags take advantage of this by making the time of a crash

as long as possible. You do not come to a sudden and abrupt halt.

The longer the time of the crash, the smaller force must become.

For this same reason, you often see barrel barricades on highway exit ramps.

Another device that makes the force smaller by making time longer is boxing gloves.

• When the time of impulse is long, the force will be small.

• When the time of impulse is small, the force will be large.

• When an object bounces it has a greater change in momentum than if the object just came to a stop.

• An object that bounces has a greater change in velocity than one that comes to a stop because it has to change direction.

The diagram to the right depicts the before- and after-collision speeds of a car which undergoes a head-on-collision with a wall. In Case A, the car bounces off the wall. In Case B, the car crumples up and sticks to the wall.

a. In which case (A or B) is the change in velocity the greatest? Explain.

b. In which case (A or B) is the change in momentum the greatest? Explain.

c. In which case (A or B) is the impulse the greatest? Explain.

d. In which case (A or B) is the force which acts upon the car the greatest (assume contact times are the same in both cases)? Explain.

• Momentum is always conserved. What does it mean for something to be

conserved?

When something is conserved, that means that its value does not change.

If you stand on ice while wearing ice skates and throw a ball out in front of you, what will be your motion?

You will move backward.

The momentum you and the ball had at the beginning has not changed.

What was the momentum of you and the ball before you threw it? (think about the equation for momentum)

Your momentum was zero before you threw the ball.

After you threw the ball you moved in a direction opposite the direction of the ball. Your momentum and the balls momentum were equal but opposite, that means that your momentum will cancel each other out.

The Law of Conservation of Momentum states that in the absence of an external force, the momentum of a system remains unchanged.

Let’s look at a sample problem.A 145 kg defensive end running at 2.5 m/s sacks a 100.0

kg quarterback at rest. After the hit, the defensive end is found to have a speed of 0.36 m/s. Determine the momentum of the defensive end before and after the tackle.

p = mvBefore: After:p = (145kg)(2.5 m/s) p = (145 kg)(0.36 m/s)p = 360 kgm/s p = 52 kg m/s

Hey, what happened!?

Wasn’t the defensive end’s momentum supposed to be conserved?

In both situations, the ice skater and the football player. There wasn’t just a single object. There were two objects interacting.

Momentum was conserved in the system.

• Momentum is conserved in a collision between two objects in an isolated system.

• An isolated system is one in which no outside forces exist.

A common example of an outside force is friction.

We will only focus on mathematical calculations where momentum is conserved, however, you need to understand the concepts of when momentum is not conserved and why.

Types of Collisions

There are two types of collisions: elastic and inelastic.

• An elastic collision is one where the two objects do not stick together.

An example of an elastic collision is two pool balls hitting each other.

• An inelastic collision is one where the objects stick together and move with the same speed after the collision.

An example of an inelastic collision is when a person on roller skates catches a thrown ball.