Projectiles Launched at an Angle

Projectiles Launched at an Angle

A refresher on projectiles that are launched straight up. Scientists over time have discovered the following relationship that represents height (I’ve also included the formula for distance, but since the projectile is launched straight up, distance isn’t very interesting):

x represents horizontal position (distance)

y represents vertical position (height)

g represents gravity (always 16 if using ft/sec, and on Earth)

vi represents initial velocity

s represents starting height

t represents time

Take note: horizontal distance (x) is 0 because the projectile is being launched straight up. But if the projectile is NOT launched straight up, things get a little more interesting. Here are the equations for height and distance:

vix represents initial velocity in the horizontal direction

viy represents initial velocity in the vertical direction

Notice that now there are two different initial velocities: one for horizontal velocity and one for vertical. Briefly, here’s why: let’s take a soccer ball kick. It is kicked with an initial velocity, but some of the velocity increases its height and some of the velocity increases its distance. How the initial velocity is split between the two depends on the launch angle. Specifically:

(theta) represents the launch angle

The above formulas for vix and viy can be substituted into the x and y functions. One other important piece to note is that there is no horizontal gravity, so g=0 in the x= equation. When you plug in all the pieces, you have an equation for height and an equation for distance, both as a function of time.

Problems

- A potato is launched across a street from a balcony 20 feet from the ground. Its initial velocity is 50 ft/sec. Where is the potato after 2 seconds? How far away is the potato when it hits the ground?
- A soccer ball is kicked at an initial velocity of 40 ft/sec and at a 30˚ launch angle. How far does it travel? How long until it hits the ground?
- An object is fired at 14 ft/sec at three different angles: 30˚, 50˚ and 70˚. Which gets the most distance? Which is in the air for the longest? If you could choose any angle to maximize distance, which would you choose? How about maximizing time in the air?