How To Calculate Catapult Force

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.

Step 1

Start by weighing your projectile. For the necessary calculations that follow, it is best to record the mass in kilograms.

Step 2

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 at such high speeds, you may want to have an assistant with a stopwatch take over 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.

Step 3

Determine the initial horizontal velocity v_h 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:

\(v_h=\frac{d}{t}\)

For example, a distance of 100 meters at 10 seconds is: v_h = 100/10 = 10 m/s.

Step 4

Determine the initial vertical velocity v_v using the time it took the projectile to reach its maximum height t_max, the acceleration due to gravity (9.8 meters/second²), and the vertical velocity at maximum height, which is zero:

\(0=v_v -gt_{max} \implies v_v=gt_{max}\)

So, if the projectile took 5 second to reach maximum height: v_v = (9.8)(5) = 49.4 m/s.

Step 5

To determine total velocity v_total based on horizontal velocity v_h and vertical velocity v_v as determined in the last two steps, we use the formula:

\(v_{total}=\sqrt{v_h^2+v_v^2}\)

Using the numbers given in previous steps we would get:

\(v_{total}=\sqrt{10^2+49.4^2}=50\text{ m/s}\)

Step 6

Next, we need to determine the acceleration of our projectile ​a​ by taking the initial velocity of the projectile v_total and dividing it by the time it took to reach that velocity t_initial. So, if t_initial is 0.25 seconds:

\(a=\frac{v_{total}}{t_{initial}}=\frac{50}{0.25}=200\text{ m/s}^2\)

Step 7

Multiply this acceleration by the mass of the projectile ​m​ and you have the amount of force exerted by the catapult ​F​ on the projectile. So if ​m​ is 1 kilogram:

\(F=ma=1\text{ kg}\times 200\text{ m/s}^2=200\text{ N}\)