Momentum

We have now looked at force and energy. We want to define a new quantity that helps connect the two: Momentum. Physically, momentum is a measurement of an object's tendency to continue in motion at a constant velocity. If the force and energy are linear, then we are talking about linear momentum. We write this mathematically as

p = mv

(13.1)

If the force and energy are rotational, then we are talking about angular momentum, which is written mathematically as

L = Iw

How is the momentum related to kinetic energy? Start with the definition of kinetic energy

(13.2)

What about force and momentum? Start with

this can be rewritten as

dp = Fdt

Impulse

Integrating both sides of the equation, we get

(13.3)

We define the right hand side of (13.3) to be the impulse, J. Physically, the impulse is the total amount of force applied to an object in order to change its momentum. From the definition, we can see that the impulse has units of Newton-seconds (N-sec).

Let's look a little more closely at the impulse. Since the right hand side is an integral, all we are really determining is the area under a graph of F verses t. However, as long as the integral remains unchanged, the specific nature of the force could be anything. Thus, we see that what we really determine when we solve (13.3) for the force is the time averaged force. This can be seen more clearly from the following picture:

Notice that in reality, the force increases very rapidly from zero to a maximum as the bat comes more into contact with the ball, and then drops back to zero as the ball rebounds off of the bat. So the force varies over the time interval of the impulse. What we calculated was a constant force, which is the second curve. The integral for each curve is the same, and represents the change in momentum.

Example:

Estimate the force exerted by the seat belt on an 80 kg driver when the car, originally moving at 25 m/s, crashes into a fixed object.

Assume that the car travels about 1 meter as the front end of the car crumples during the collision. This is also the distance traveled by the driver during the collision if he is wearing a seat belt. Also assume that the acceleration of the car is uniform as it crashes. Then the average speed of the car during the collision is one-half the initial speed, or . The time of collision is then

The total impulse received is

so the average force is

Conservation of Linear Momentum

Next, we turn to another important conservation law: Conservation of Linear Momentum. From conservation of energy, we saw that for an isolated system, the amount of energy in the system must remain constant. However, since energy is just a number, it does not allow us to solve for how the vector quantities, such as force, are distributed. For this we need conservation of linear momentum. Physically, it can be stated as when the vector sum of the external forces acting on a system is zero, the total linear momentum of the system remains constant, even though the distribution of momentum in the system might be different. Mathematically, we can write this as

d(Sp) = 0

(13.4)

Example:

You are playing a game of pool and there is only the 8 ball left on the table. If you hit the cue ball with a force of 50 N for 0.001 sec, what is the speed of the 8 ball when it goes into the pocket? Assume that there is no friction, that the balls have a mass of 0.025 kg, and the cue ball has a final speed of 1.5 m/sec at an angle of 30 degrees from its original path.

1) Determine the change in momentum to the cue ball.

From the previous section, we know that the change in momentum is equal to the impulse applied. Assume that the force is applied in the x direction, then

2) Determine the change in momentum of the 8 ball.

We can treat the cue ball and the 8 ball as an isolated system with no external forces acting on them. Then, if we assume that the 8 ball is at rest before the interaction, from conservation of linear momentum we have

Sp_after - Sp_before = 0

where

Thus,

Solving for p_8,a, get

3) Since there is no friction, by conservation of linear momentum the 8 ball enters the pocket with the same momentum it had when it started rolling. So the final speed of the ball is just

The relation between the conservation of linear momentum and the conservation of energy is stronger than it first appears. With the understanding that time and space are really connected into a single object, the two conservation principles become united into a new single conservation law, conservation of 4-momentum.