Momentum and Impulse

What is momentum?

Momentum has a mathematical definition.

momemtum = mass × velocity

In symbols this becomes

p = m × v

where momentum (p) has the units of units . Momentum is a vector.

In common English, we tend to confuse momentum with inertia. (Recall that intertia is a measure of an object's resistance to change in motion, measured in kilograms.) However there is a distinct difference between the two. Momentum depends upon velocity, where inertia is simply mass. An object at rest still has inertia but has no momentum.

Example 1 - Bullet vs. Car

Which has more momentum, a 50.0g bullet moving at 325m/s, or a 2250kg Toyota moving at 5.15m/s?

Newton's 2nd Law of Motion

Newton first stated his 2^nd law of motion (what we now call F=ma) in terms of momentum. He stated that a force F applied during a time interval t was required in order to change something's momentum p.

acceleration

Some simple algebra will convert this to the familiar 2^nd law formula more commonly used today.

f=ma

Example 2 - Slowing a train

What force is required to bring a 650,000kg freight train to rest from a speed of 25m/s in a time of 15 minutes?

The Impulse-Momentum Theorem

A quick rearrangement of the 2^nd law formula in terms of momentum gives us the equation:

F × t = Δp

The combination of force and time (measured in N s) is a quantity called impulse. An impulse is a force F applied during a time period t that causes a change in momentum.

F × t = Δp

impulse = change in momentum

Example 3 - Wrecking the Ferrari

Moustapha Jones is in two car wrecks.
A) While not wearing his seatbelt, Moustapha (75kg) crashes at 25m/s into a concrete wall. It takes his steering wheel 0.025s to bring him to rest. What average force did the steering wheel apply on his chest?

B) In the 2nd wreck (same speed, same mass) he is wearing his seatbelt and is brought to rest in 0.25s. What force was provided by the seatbelt?

Moustapha Jones is in two car wrecks.
B) In the 2nd wreck (same speed, same mass) he is wearing his seatbelt and is brought to rest in 0.25s. What force was provided by the seatbelt?

Vectors Again!

Momentum and impulse are both vectors, and as such you should consider their direction when doing calculations.

Example 4 - Space Cowboy

An astronaut (150kg) is floating in space at 5.0m/s. He fires a thrust from his jetpack of 55N for 2.2s at 90° to his direction of travel. Find his new speed.

Graphing Impulse

Since an impulse is a force that is spread out over time, it is common to use a graph to show how this force is related to time. This can be done on a force-time graph as shown in the example below.

impulse graph

On a force-time graph, the area under the curve represents the impulse provided by the force. This impulse can cause a change in momentum.

Example 5- Serving the Ace

A tennis ball (m=60g) is served using a force described in the graph below.
A) Find the total impulse given to the ball.
B) Find the final speed of the ball, assuming the ball was at rest when served.