Momentum and Collisions

Newton’s 1st Law of Motion

“An object at rest will stay at rest, and an object in motion will stay in motion unless acted upon by a net force.”

Also known as the Law of Inertia.

Inertia

Inertia is the tendency of an object to resist changes in motion.

An object’s mass determines its inertia.

Momentum

The linear momentum of an object of mass m moving with velocity v is defined as the product of the mass and the velocity. Momentum is represented by the symbol p.

p = momentum of object (kg m/s)

m = mass of object ( kg)

v = velocity of object (m/s)

p = m v

Momentum

The SI unit of momentum is kg m/s.

Sample Problem

Determine the momentum of a 60-kg halfback moving eastward at 9 m/s.

Sample Problem

Givens:m = 60 kgv = 9 m/s

Formula:p = m v

Solution:

p = (60 kg)(9 m/s) = 540 kg m/s

The momentum of the halfback is 540 kg m/s.

Solve These Problems:

1. What is the momentum of a 1000 kg car travelling at 25 m/s?

2. What is the momentum of a 250 kg skater moving at 6 m/s?

3. What is the velocity of a 5000 kg truck that has momentum of 100,000 kg m/s?

25,000 kg m/s

1500 kg m/s

20 m/s

Newton’s 2nd Law of Motion

The Law of Acceleration

F = ma

Impulse

Newton’s 2nd Law of Motion as it relates to momentum becomes:

where FΔt is the impulse and Δp is the change in linear momentum.

ptF Δ=Δ

Impulse

In simple terms, a small force acting for a long time can produce the same change in momentum as a large force acting for a short time.

ptF

Effect of Collision Time Upon the Force

Racket and Bat SportsThe act of following through

when hitting a ball increases the time of collision and contributes to an increase in the velocity change of the ball.

In tennis, baseball, racket ball, etc., giving the ball a high velocity often leads to greater success.

Newton’s 3rd Law of Motion

“For every action there is an equal and opposite reaction.”

Consider a pair of objects interacting with each other.

According to Newton’s 3rd Law, the forces they exert on each other must be equal and opposite.

Conservation of Momentum

In every interaction between two isolated objects, the change in momentum of the first object is equal to and opposite to the change in momentum of the second object.

Thus in all interactions between two isolated objects, momentum is conserved.

Law of Conservation of Momentum

To solve conservation of momentum problems, use the formula:

The sum of the momenta before the collision equals the sum of the momenta after the collision.

afterbefore pp

Total Momentum

The momentum of each object could change before and after an interaction, but the total momentum of the two objects together remains constant.

Three Types of Problems

Explosion Both objects at rest before the collision Objects move off in opposite directions

Two objects in motion – Elastic Collision Objects have initial velocity (one may be at rest) After collision objects have different velocities

Objects move together - Inelastic Collision One or more objects in motion before collision After collision, objects move together with same

velocity (This could be reversed.)

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p = momentum before collision (kg m/s)

p’ = momentum after collision (kg m/s)

m1 = mass of object 1 (kg)

v1 = velocity of object 1 before the collision (m/s)

m2 = mass of object 2 (kg)

v2 = velocity of object 2 before the collision (m/s)

v1’ = velocity of object 1 after the collision (m/s)

v2’ = velocity of object 2 after the collision (m/s)

Head-On Collision

Car “Rear Ends” Truck

Truck “Rear-Ends” Car

The Cart and the Brick

Sample Problem

Two skaters stand facing each other. One skater’s mass is 60 kg, and the other’s mass is 72 kg. If the skaters push away from each other without spinning,

a. the 60 kg skater travels at a lower momentum.

b. their momenta are equal but opposite. c. their total momentum doubles. d. their total momentum decreases.