SfC Home > Physics > Motion >

Applications of Collision Equations

by Ron Kurtus (revised 2 January 2014)

The general equations for the collision of a moving object A with a stationary object B was determined in Derivation of a Simple Collision and are stated below.

Applications of the equation include when the mass of A < B, A = B, and A > B.

Some questions you may have include:

• What is the collision equation?
• What is the solution when mass of A is less than B?
• What is the solution when mass of A equals that of B?
• What is the solution when mass of A is greater than B?

This lesson will answer those questions. Useful tool: Units Conversion

Collision equations

The relationships between the velocities when object A collides with object B are:

Of v2 and V2 as a function of v1 are:

v2 = v1(1 − k)/(1 + k)

V2 = 2v1/(1 + k)

Between v2 or V2:

v2 = V2(1 − k)/2

V2 = 2v2/(1 − k)

where

• v1 is the initial velocity of object A
• v2 is the resulting velocity of object A
• V2 is the resulting velocity of object B
• k is the ratio of the masses of the objects: k = M/m
• m is the mass of object A
• M is the mass of object B

Mass of A is less than B

Consider the situation when the mass of A < B. Suppose M = 2m. Then k = 2.

v2 as a function of v1

v2 = v1(1 − k)/(1 + k)

v2 = v1(1 − 2)/(1 + 2)

v2 = −v1/3

V2 = 2v1/(1 + k)

V2 = 2v1/(1 +2)

V2 = 2v1/3

Relationship of v2 and V2

v2 = V2(1 − k)/2

v2 = V2(1 − 2)/2

v2 = −V2/2

This means that for a case when M = 2m, object A would move in the opposite direction in a velocity of 1/2 the velocity of object B.

Resulting motion when mass of A less than B

Note: Exactly what happens at the point of collision is not considered in this derivation.

Mass of A equals B

Suppose the mass of object A is the same as that of object B. Then k = 1.

v2 = V2(1 − k)/2

v2 = V2(1 − 1)/2

v2 = 0

However, to maintain the conservation of momentum and energy, you can use the momentum equation:

v1 = v2 + kV2

v1 = V2

Thus, the collision sequence looks like:

Collision of equal mass objects

This means that when the masses of A and B are equal, the collision results in object A becomeing stational and object B moving foward at the same velocity as v1. This effect can be seen in Newton's Cradle.

Mass of A is greater than B

Suppose m = 2M (or M = m/2). Then k = 1/2.

v2 = V2(1 − k)/2

v2 = V2(1 − 1/2)/2

v2 = V2(1/2)/2

v2 = V2/4

Also, since

V2 = 2v1/(1 + k)

V2 = 2v1/(1 + ½)

V2 = 2v1/(3/2)

V2 = 4v1/3

and

v2 = v1/3

The collision scenario is:

Resulting motion when mass of A greater than B

This means that for a case when m = 2M, object A would continue to move in the same direction after the collision at a velocity of 1/4 the velocity of object B.

Summary

By inserting the value of k in the collsion equations, you can see the resulting motion of the objects.

Resources and references

Ron Kurtus' Credentials

Websites

Physics Resources

Books

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

Students and researchers

www.school-for-champions.com/science/
motion_collision_applications.htm

Where are you now?

School for Champions

Physics topics

Motion topics

Let's make the world a better place

Be the best that you can be.

Use your knowledge and skills to help others succeed.

Don't be wasteful; protect our environment.

Live Your Life as a Champion:

Seek knowledge and gain skills

Do excellent work

Be valuable to others

Have utmost character

Be a Champion!

The School for Champions helps you become the type of person who can be called a Champion.