  Pendulum Physics

This page is reproduced from "Backyard Ballistics" by permission of William Gurstelle.

Pendulum Physics

Earlier in this book, Sir Isaac Newton's three laws of motion were discussed. Let's apply those laws of motion to determine the velocity of a bullet.

Physicists would say that the collision between the spud and the pendulum is "perfectly inelastic" because the wadded-up newspapers allow for no bounce or rebounding whatsoever when the potato hits the pendulum. In this type of collision, physicists would say that "momentum is conserve." However, in this type of collision, where there is no bounce or elasticity between the colliding objects, those same scientists would note that the kinetic energy is not conserved.

Energy and mass conservation laws are the basis of anlysis that scientists use to relate mass, velocity, distance and time. Through careful choice of the variables involved, and mathematical manipulation of the physics equations that describe the process, an equation can be written to let us determine the speed of the projectile from the easily measured variables discussed earler.

V potato = (1 + M/m) sqrt(2gh)

It is beyond the scope of this book to derive the equations for muzzle velocity. Most beginning physics textbooks discuss the physics of the ballistic pendulum when one-dimensional particle kinetics and energy and momentum conservation laws are introduced. For now, take it on faith that the muzzle velocity of the potato is given by this equation.

 where M is the weight of the pendulum, m is the weight of the potato, g is the earth's gravitational constant, and h is the vertical rise of the pendulum.

If we insert the constants, we can make a fairly easy-to-use equation to determine the speed of any projectile shot into our pendulum:

V potato = (1 + M/m) sqrt(43.9 x h)

• The velocity of projectile is in miles per hour.
• The weight of projectile and pendulum can be ounces, pounds, or grams, but units must be used consistently.
• The height of pendulum rise is in feet (or use inches and divide by 12).

To find the muzzle velocity of any projectile, substitute your figures in the equation given.

Tips and Troubleshooting

1. A weight of four pounds works well for the pendulum when measuring muzzle velocity of a potato cannon. You can use lead fishing weights in a small bag or something similar to make the box weigh as close to four pounds (64 ounces) as possible.
2. To get an accurate reading of the muzzle velocity, it is very important that the pendulum and potato be measured very accurately. Use a postal scale to determine weights to a half-ounce or better.
3. Take care to line up the marker exactly perpendicular to the paper. The marker tip should rest squarely and lightly on the paper.
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