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?

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 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

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.

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