Probably one of the most famous, or infamous, siege weapons -- the catapult was used to cast projectiles into an enemy stronghold in an attempt to either weaken its defenses or break the will of those sheltered inside. From a physics point-of-view, the catapult is actually a simple lever, with the catapult arm pivoting on a fulcrum until a crossbar stops the arm and releases the projectile sitting in the bucket at the end of the arm. If you have access to a catapult or make a simple one -- determining the force of it only requires a few measurements and some simple calculations.
Determining the Force of Your Catapult
Start by weighing your projectile. For the necessary calculations that follow, it is best to record the mass in kilograms.
Before launching your projectile, get in position to measure: how long it takes the arm to travel from resting position to hitting the crossbar, how long the projectile takes to reach maximum height, how far the projectile travels and how long it takes to reach impact. Since the catapult moves a such high speeds -- you may want to have an assistant with a stopwatch takeover one of the time measurement, use a video camera to capture the catapult in action and take measurements based on the video footage, or use multiple trials to get all of your data points.
Determine the initial horizontal velocity (Vh) using the projectile's impact distance (d) and the length of time it took to get there, assuming that the projectile was traveling at the same horizontal speed upon impact: (Th): Vh = d/Th. For example, a distance of 100 meters at 10 seconds is: Vh = 100/10 = 10 meters per second.
Determine the initial vertical velocity (Vv) using the time it took the projectile to reach its maximum height (Tmax), the acceleration of gravity (-9.8 meters/second^2) and the vertical velocity at maximum height, which is zero: Vv= 0 - (gravity * Tmax). So, if the projectile took 5 second to reach maximum height: Vv = 0 - (-9.8 * 5 seconds) = 49.4 meters/second.
To determine total velocity (Vtotal) based on horizontal velocity (Vh) and vertical velocity (Vv) as determined in the last two steps, we use the formula: Vtotal = the square root of (Vv squared + Vh squared). Using the numbers given in previous steps we would get: Vtotal = the square root of (10^2 + 49.4^2) = the square root of (100 + 2440) = about 50 meters/second.
Next, we need to determine the acceleration of our projectile (Aproj) by taking the initial velocity of the projectile (Vinitial) and dividing it by the time it took to reach that velocity (Tinitial). So, if Tinitial is 0.25 seconds: Aproj = Vinitial/Tinitial = 50/0.25 = 200 meters/second^2.
Multiply this acceleration (Aproj) by the mass of the projectile (Mproj) and you have the amount of force exerted by the catapult (Fcat) on the projectile. So if Mproj is 1 kilogram: Fcat = Mproj x Aproj = 1 x 200 = 200 kg*m/second^2 = 200 Newtons (a standard unit of force).
TL;DR (Too Long; Didn't Read)
It's best to use a metric tape measure to record your data since the simplest figure for the acceleration of gravity (-9.8 meters/second^2) is in metric.
Before firing your catapult, make sure that it will not cause damage to people or property.