The Math Behind A Catapult Siege Weaponry, Castle Battles, or just firing breakfast cereal off the table at your little sister using a spoon-­‐-­‐-­‐we’ve all seen catapults. But WHY do they work exactly? And how can we build one to hit a certain target? Well, I’m glad you asked. Let’s take a look. Catapults operate off the principle that gravity pulls downward, and if we can fling something with enough force we can defy gravity and travel upward-­‐and-­‐outward to achieve great heights. However, if you’ve ever fling something using a catapult, you’ll know that things don’t just go up, they also come back down. That’s because even though we are flinging something into the air, we have that object’s weight, as well as the continually acting force of gravity, wind, initial velocity, and other factors that tend to act on our upward flinging to create a curved motion. The flight of an object from a catapult looks like this: There are several problems and steps to derive the formula I’m about to show you, but suffice to say, if another mathematician has already done the derivation, just use the formula. You have bigger questions to answer. SO here is the general equation for vertical motion of a projectile on the earth's surface. d = vt sin(x) - 4.9t^2
Where d represents distance traveled, v is initial velocity, t is time,
and x represents the angle at which you launch the catapult.
Changing any of these variables will change where, how far, and how
fast, the object flies off a catapult.
Let's say we have a projectile being fired from your catapult at 4m/s
at 45 degrees from the ground. We can use math to work out everything
we need to know.
Laws of Trigonometry tell us that we can relate the initial velocity to the ground and maximum height of the projectile using a triangle: Using this triangle and what we know about trig functions, we can express vertical and horizontal motion using these formulas: (Which is useful because later we plug them back into the vertical motion equation we found previously). Keep reading J Sin(x) = b/v cos(x) = a/v Where a represents the “over” part of the “up-­‐and-­‐over” that happens when you launch a projectile. Except in “math terms’ We call that the horizontal component of movement. Meaning, when you throw it, the object doesn’t just go up, it also goes “over” (That’s what allows you to hit things in front of you). So we can solve this cosine equation for “a” to find out how to express the horizontal component of our toss. A = v cos(x) We do the same thing with the sine equation, only this time we solve for b, which is the “up” part of our toss. We call that the vertical component. B = v sin(x) These two equations represent how fast “up” and how fast “over” the object is traveling. Since we know Distance = Rate(Time), we can note that Rate = Distance/Time SO in tehse cases, horizontal motion = distance traveled/time Vertical motion = distance traveled/time So if we can solve both equations for d, we will be able to come up with one equation that describes the whole situation. We start with the horizontal components and we solve for distance traveled. (Which is a mathy way of saying, we can figure out how far the object is going to go if we know how fast we are throwing it and at what angle) We know that we threw it at a speed of 4 m/s (I told you that part) , and we also know that we fired it from an angle of 45 degrees. So we plug this information into our equation for horizontal movement. A = (4)cos(40) = distance/time 3.06(time) = distance Then since we know the equation for vertical motion (First page), we can plug the above information into that equation here: 3.06t = 2.57t – 4.9t^2 The reason we do not use the equation above where we solved for B is because we want to end up with one equation and one variable. So we choose substitution instead. Once we have this place, we can use the quadratic formula to solve and find out how long the object will be in the air (solve for t). You can use the equations to answer many questions about the flight of an object launched by catapult. Even though this catapult is relatively simple, NASA engineers, and other professionals use this same kind of math every day when they are calculating shuttle launches, monitoring national security, or even just repairing an elevator or practicing archery. There are many real-­‐world applications for Trigonometry, just look around. What applications can you find?