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Projectiles Launched at an Angle

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Author: Erin Bjornsson

Throwing a ball to a friend or shooting a cannon are both examples of projectile motion. Gravity is a force that acts upon objects, drawing them towards the center of the Earth at 9.81 m/s2. Horizontal motion happens when an object is acted upon by an outside force, and it will stay in motion until acted upon by another force, including hitting the ground. Newton’s Third Law of Motion says that an object will stay in motion unless acted on by an outside force, so this means that there is no acceleration in the horizontal direction.

The angle at which something is thrown or shot also affects how far it will travel, because what goes up must come down!

Problem: Find the launch angle of a projectile for the longest distance.

Which angle will launch the projectile the farthest? Why? What happens if the launch angle is smaller? Greater?

Materials

  • Nerf gun and darts
  • Tape measure
  • Calculator

Procedure

  1. Hold out your arm at a 90° angle.
  2. Use the tape measure to measure from your hand to the floor vertically. Record this height in meters.
  3. Use the following formula to calculate the time it would take for an object to fall from this height:

t = (2h / g)½

Where t is time in seconds, 

h is the height,

and g is the acceleration due to gravity.

  1. Fire the Nerf gun horizontally and measure the distance it travels from where it was shot. Be sure to measure from where it first hits the ground.
  2. Calculate the firing velocity.

v = d / t

Where v is the velocity in meters per second,

d is the distance traveled, in meters,

and t is the time you calculated in step 3.

  1. Experiment with firing the Nerf gun at different angles. Use the following equation and predict the distance it will travel. Calculate the height of each trial as well. Try to hit a target!

d = (v2 / g)sin(2θ)

Where d is the distance traveled, in meters

v is the firing velocity, in meters/second

g is the acceleration due to gravity and θ is the measure of the firing angle, measured from the horizontal.

  1. You can also turn the equation around to find maximum height:

h = (vsinθ)2 / 2g

Results

The 45° launch angle will go the farthest distance.

Why?

The 45° angle goes the farthest because it provides both a long time for the dart to be in the air and constant enough horizontal speed.

The steeper the angle gets, the greater the vertical component of the velocity is, which means more time in the air. However, this means the horizontal component of the velocity is small, so the dart doesn’t cover much distance. When the measure of the angle is small than 45°, the horizontal component of the velocity is large, but the dart doesn’t spend much time in the air because it starts off closer to the ground. 

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