Conceptual Physics

7.6 - Derivation: conservation of momentum from Newton’s laws

Newton formulated many of his laws concerning motion using the concept of momentum, although today his laws are stated in terms of force

and acceleration. The law of conservation of momentum can be derived from his second and third laws.

The derivation uses a collision between two balls of masses m 1 and m 2 with velocities v 1 and v 2. You see the collision illustrated above, along

with the conservation of momentum equation we will prove for this situation.

To derive the equation, we consider the forces on the balls during their collision. During the time ǻt of the collision, the balls exert forces F 1

and F 2 on each other. (F 1 is the force on ball 1 and F 2 the force on ball 2.)

Diagram

This diagram shows the forces on the balls during the collision.

Variables

Strategy

1. Use Newton’s third law: The forces will be equal but opposite.


2. Use Newton’s second law, F = ma, to determine the acceleration of the balls.


3. Use the definition that expresses acceleration in terms of change in velocity. This will result in an equation that contains momentum (
   mv) terms.

Physics principles and equations

In addition to Newton’s laws cited above, we will use the definition of acceleration.

a = ǻv/ǻt

Conservation of momentum

m 1 vi1 + m 2 vi2 = m 1 vf1 + m 2 vf2

p = mv = momentum

m 1 ,m 2 = masses of objects

vi1,vi2 = initial velocities

vf1,vf2 = final velocities

duration of collision ǻt

ball 1 ball 2

force on ball F 1 F 2

mass m 1 m 2

acceleration a 1 a 2

initial velocity vi1 vi2

final velocity vf1 vf2

(^150) Copyright 2007 Kinetic Books Co. Chapter 07