How to Find Maximum Height of a Projectile
Ever wanted to know how high and how fast you can kick a football? Turns out, it’s pretty easy to figure out. All you need to know is how far away from you the ball lands and how much time it spends in the air (the “hang time”). Combine these two simple measurements with a little bit of physics and you can start competing with your friends to see who can kick the highest or the fastest!
Problem: How can you find a projectile's maximum height?
Given the hang time and distance traveled of a football, figure out how high it went, how fast you kicked it, and the angle at which it left the ground.
- Tape measure or a football field
- Pencil and paper
- Willing friend
- Kick the ball and, at the same time, start the stopwatch. This is where a friend can come in handy. Have them hold the ball and work the timer so you can focus all your energy on giving the ball the best kick you can!
- When the ball hits the ground, stop the timer and mark where the ball first lands. Measure how far away that is from where you kicked it. A football field makes this pretty easy, but a long tape measure works just as well.
- Now for a bit of math. Divide the distance traveled (d, meters) by the amount of hang time (t, seconds). This tells you the ball’s horizontal speed, vx, in meters per second: vx = d / t.
- Calculate the ball’s vertical speed, vy, by multiplying half the hang time (t) by the acceleration due to gravity (g = 9.8 m/s2): vx2 + vy2 = ½ gt
- Now it’s time to combine the horizontal and vertical speeds to get the total speed, v = √(vx2 + vy2). That’s how fast you kicked the ball, in meters per second.
- To calculate how high the ball went (h), take the vertical velocity squared and divide it by twice the gravitational acceleration: h = vy2/ 2g
- You can also calculate the angle at which the ball left the ground (θ) by using a tiny bit of trigonometry: θ = tan-1(vy / vx)
You have calculated the speed at which you kicked the ball (in meters/second), the angle at which it launched, and how high you kicked it (in meters).
The above may sound like a bunch of mathematical gobbledygook. But it’s based on a very simple, and very important, idea from physics: you can treat the vertical and horizontal motion of the ball independently.
The total time spent in the air combined with how far along the ground the ball went tells you everything you need to know about the ball’s horizontal velocity. Ignoring air resistance, the ball doesn’t experience any horizontal acceleration, so its horizontal velocity stays constant.
The ball’s vertical motion is a different story. As soon it leaves your foot, gravity starts slowing the ball down. Eventually, the ball’s vertical velocity reaches zero. After that, the ball turns around and starts falling back to Earth, picking up speed the entire time. Ignoring air resistance (again!), the ball’s vertical speed when it hits the ground is the same as its vertical speed when you kicked it.
Since the final and initial vertical speeds are the same, we can focus on just the second half of the ball’s trip. We can ask, how fast would a ball be moving after a certain amount of time if you dropped it from a great height? That’s another way of asking how fast the ball is traveling when it hits the ground after falling from the highest part of its journey. The “falling time” is half the hang time.
The total velocity comes from combining the horizontal and vertical velocities. We can draw the velocities like a right triangle. The horizontal and vertical velocities make up the sides of the triangle while the total velocity is its hypotenuse. Using the Pythagorean Theorem, you can use the sides to figure out the total speed with which the ball was launched. You can use the same triangle to figure out the angle at which it took off.
Figuring out the height comes back to just worrying about the ball’s vertical motion. We know how quickly the ball left your foot. And we know how strongly gravity is working to slow it down. That’s all we need to figure out how high the ball went. It’s the same as knowing how far your car will go if you’re driving at 60mph and suddenly hit the brakes. Except, in this case, the brakes are gravity!
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