One-dimensional

Elastic and Inelastic Collisions

Of course, collisions come in many forms. We will restrict our attention to two very special kinds.

Inelastic collisions are those in which the two objects stick together.

Consider a situation with mass m₁ coming in with initial velocity v_1i and striking mass m₂ that is at rest (v_2i = 0).

The two masses stick together so this is a totally inelastic collision.

The initial momentum is given by

P_Tot,i = p_1i + p_2i

P_Tot,i = m₁ v_1i

After the collision, the final momentum is given by

P_Tot,f = (m₁ + m₂) v_f

Since momentum is conserved, this gives

P_Tot,f = P_Tot,i

(m₁ + m₂) v_f = m₁ v_1i

v_f = [ m₁ / (m₁ + m₂) ] v_1i

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Elastic collisions are those in which the Kinetic Energy is also conserved.

Consider two objects that "interact" or collide or have some effect on each other -- in such a way that the total Kinetic Energy is also conserved. This is a totally elastic collision.

P_Tot,i = P_Tot,f

m₁ v_1i + m₂ v_2i = m₁ v_1f + m₂ v_2f

Remember, momentum is a vector! That means that a velocity to the right is positive and a velocity to the left is negative.

Since this is an elastic collision, we also have

K _Tot,i = K _Tot,f

(¹/₂) m₁ v_1i² + (¹/₂) m₂ v_2i ² = (¹/₂) m₁ v_1f² + (¹/₂) m₂ v_2f ²

This provides two equations from which we can solve for the two unknowns, v_1f and v_2f .

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A special case that is interesting to look at is that of an "incoming" object of mass m₁ with initial velocity v_1i that collides elastically with mass m₂ that is at rest. That is v_2i = 0.

The equations for the final velocities are

and

Notice what happens when the masses are equal; for m₁ = m₂:

And notice for m₁ > m₂:

And notice for m₁ < m₂:

These elastic collisions are consistent with -- they are predicted by -- these equations:

and

I have some animation for these collisions on my web site for PHY 3050 which I have not yet incorporated into this web site. Click here for that animation.

PHY 3050 has a lighter description of SPLAT - BOING! problems that you might enjoy.

Collisions			2D Collisions
Return to ToC, Ch9, Linear Momentum

(c) Doug Davis, 2001; all rights reserved