Mechanics Lecture 19: Momentum, Impulse, and Conservation of Linear Momentum, Lecture notes of Physics

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Physics 170 - Mechanics

Lecture 19

Momentum & Impulse

Conservation of Linear Momentum

Linear Momentum

Momentum is a vector; its direction is the

same as the direction of the velocity.

There is a quantity which remain the same

even when objects crumple or fold, it’s the

linear momentum.

Momentum &

Newton’s Second Law

Newton’s second law, as we wrote it before, is: is only valid for objects that have constant mass. Here is a more general form in terms of momentum, which is also useful when the mass is changing:

Problem Solving Strategy Momentum Picture: Determine that the net external force Σ F ext (or Σ F ext x ) on the system is negligible for some time interval. Solve:

1. Draw a sketch showing the system before and after the time interval. Include coordinate axes and label the initial and final velocity vectors.
2. Equate the initial momentum to the final momentum and express this as a vector equation (or one or more scalar equations involving x, y, and z components.)
3. Substitute the given information into the equation(s) and solve for the quantity or quantities of interest. Check: Make sure you include any minus signs that accompany velocity components, because momentum can have either sign.

Impulse

Impulse is a vector, in the same direction

as the average force.

A large impulse causes a big change in a

object’s momentum

Impulse

We can rewrite

as

So we see that

The impulse I is equal to the change in

momentum Δ p.

Impulse and Average Force

Example: Hitting a Baseball (1)

A 150 g baseball is thrown at a speed of 20 m/s. It is hit straight back to the pitcher at a speed of 40 m/s. The interaction force is as shown here. What is the maximum force Fmax that the bat exerts on the ball? What is the average force Fav that the bat exerts on the ball?

Problem Solving Strategy Impulse Picture: To estimate the average force Fav , we first estimate the impulse I of the force. Assuming other forces are negligible, the impulse of the force is the net impulse, which is equal to the change in momentum, i.e., the mass times the change in velocity. An estimate of the velocity change Δ v can be made from estimates of the collision time Δ t and displacement Δ r. Solve:

1. Calculate (or estimate) the impulse I and the time Δ t. This estimate assumes that during the collision, the collision force is very large compared to all other forces on the object. This procedure works only if the displacement during collision can be determined.
2. Draw a sketch showing before and after positions of the object. Add coordinate axes and label velocities and displacement.
3. Calculate the momentum change during the collision. (I= Δ p=m Δ v)
4. Use Fav=I/ Δ t to calculate the average force. Check: Average force is a vector, and should be in the same direction as Δ v.

Conservation

of Linear Momentum

Momentum:

Conservation of Momentum