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Calculate the Range of a Projectile Fired at an Angle

If you fire a projectile at an angle, you can use physics to calculate how far it will travel. When you calculate projectile motion, you need to separate out the horizontal and vertical components of the motion. This is because the force of gravity only acts on the projectile in the vertical direction, and the horizontal component of the trajectory’s velocity remains uniform.

Shooting a cannon at a particular angle with respect to the ground.
Shooting a cannon at a particular angle with respect to the ground.

Here’s an example: Imagine that you fire a cannonball at an angle, as shown in the preceding figure. Given the initial speed of the cannonball and the angle at which it was shot, can you determine how far it will travel?

How do you handle the motion of an object shot up at an angle? Because you can always break motion in two dimensions into x and y components, and because gravity acts only in the y component, your job is easy. All you have to do is break the initial velocity into x and y components:

image1.png

These velocity components are independent, and gravity acts only in the y direction, which means that vx is constant; only vy changes with time, using the following equation:

vy = vyi + at, or

image2.png

If you want to know the x and y positions of the cannonball at any time, you can easily find them. You know that x is

image3.png

And because gravity accelerates the cannonball vertically, here’s what y looks like (the t2 here is what gives the cannonball’s trajectory in the figure its parabolic shape):

image4.png

You figure out the time it takes a cannonball to hit the ground when shot straight up (ignoring air resistance) like this:

image5.png

Knowing the time allows you to find the range of the cannon in the x direction:

image6.png

So there you have it — now you can figure out the range of the cannon given the speed of the cannonball and the angle at which it was shot.

Here’s an example: What’s the range for your cannon if you aim it at 45 degrees, which gives you your maximum range? If the cannonball has an initial velocity of 860 meters/second, the equation you use looks like this:

image7.png

Your range is 75 kilometers, or nearly 47 miles. Not bad.

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