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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:

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

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 as a function of v1

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

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

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

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.


Test your ideas


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