How to Find the Velocity of an Object before Collision

In physics, the principle of conservation of momentum comes in handy when you can’t measure velocity with a simple stopwatch. Say, for example, that you accept a consulting job from an ammunition manufacturer that wants to measure the muzzle velocity of its new bullets. No employee has been able to measure the velocity yet, because no stopwatch is fast enough. What do you do? You decide to arrange the setup shown in the figure, where you fire a bullet of mass m₁ into a hanging wooden block of mass m₂.

Shooting a wooden block on a string allows you to experiment with velocity, but don’t try this at home!

The directors of the ammunition company are perplexed — how can your setup help? Each time you fire a bullet into a hanging wooden block, the bullet kicks the block into the air. So what? You decide they need a lesson on the principle of conservation of momentum. The original momentum, 
you explain, is the momentum of the bullet:

p_i = mv_i

Because the bullet sticks in the wooden block, the final momentum is the product of the total mass, m₁ + m₂, and the final velocity of the bullet/wooden block combination:

p_f = (m₁ + m₂)v_f

Because of the principle of conservation of momentum, you can say that

p_f = p_i

Therefore, you can plug in the earlier expressions for final and initial momentum:

The directors start to get dizzy, so you explain how the initial kinetic energy of the block after it’s struck (with the bullet lodged inside it) goes into its final potential energy when it rises to height h. Here’s how you can represent the block-plus-bullet’s initial kinetic energy and the bullet-and-block’s change in potential energy:

You can plug in the value of v_f, which gives you

With a flourish, you explain that solving for v_i gives you the bullet’s initial velocity:

This should be simplified to:

You measure that the bullet has a mass of 50 grams, that the wooden block has a mass of 10.0 kilograms, and that upon impact, the block rises 50.0 centimeters into the air. Plugging in those values gives you your result:

The initial velocity is 630 meters per second, which converts to about 2,070 feet per second. “Brilliant!” the directors cry as they hand you a big check.