Analysis: Impulse-momentum

The coins in the shove ha’penny game definitely accelerated. We’ll find out that they were subjected to an impulse that changed their momentum. These words all started with ideas from Sir Isaac Newton. In his famous second law of motion, Sir Isaac explained that forces produce acceleration.

Law two, the change of motion is proportional to the motive force impressed, and is made in the direction of the right line in which that forces is impressed. That’s Motte’s translation revised by Cajori. Even after translation from the Latin, his words are unfamiliar to us now, but the concepts carry over. Motion is, mass times velocity. Change of motion would be mv2 minus mv1. And motive force impressed would be force times time. So f times t equals mv2 minus mv1, where the change in v is in the same direction as F. Here’s a simple example. It has straight line motion and constant force. If it didn’t, we’d need calculus.

The math starts at speed v1, and is acted on by a constant force F. t seconds later, it is traveling at speed v2. We can describe this by F times t equals mv2 minus mv1. We call this the impulse-momentum relationship. Now, we’ll see how it relates to acceleration. mv is momentum, Ft is an impulse. Because we specified that it all happens in a straight line, and the force is constant, we can rearrange this as F equals m times v2 minus v1 all divided by the time t. Remember, if the force or direction varies, we need to use calculus, but for our conditions, this is not necessary.

We can write v2 minus v1 all divided by t is the rate of change in velocity with time, which is acceleration a. So we get F equals ma, a very famous equation, often called Newton’s second law. Now, how can this help us understand shove ha’penny. Well, the shove the player gives a coin can be described as an impulse, that’s force times time. This creates a change in velocity from zero. Friction on the board will bring the coin to a halt, just like rolling resistance stopped the toy tractor.

We could calculate this from F equals ma, and a few other equations that relate acceleration, velocity, and displacement. A skilled player will calibrate the impulse to bring the corn between the required lines. A player might cause one coin to hit another, to move it into a scoring position. Impulse-momentum can analyse this too. If it is a direct hit, it is called direct central impact. If it is a glancing hit, it is called oblique central impact.

We won’t take this any further here, but if you would like to learn more about it, see the adaptive tutorial on impulse-momentum. And if you can’t wait to learn about acceleration and Newton’s second law, try the adaptive tutorial on projectile motion. Then it will be time for you to review where we’ve come from.