Projectile Motion Range Calculator

What is Projectile Motion?

Projectile motion is a form of motion in which an object is launched or thrown into the air and follows a curved path under the influence of gravity. The range of a projectile is the horizontal distance it travels before returning to its initial height.

Formula

The formula for the range of a projectile is:

$$R=\frac{v_0^2\sin (2\theta )}{g}$$

Where:

• $R$ is the range (in meters, m)
• $v_0$ is the initial velocity (in meters per second, m/s)
• $\theta$ is the launch angle (in radians)
• $g$ is the acceleration due to gravity (approximately 9.81 m/s²)

Calculation Steps

Let's calculate the range for a projectile:

1. Given:
   • Initial velocity ($v_0$) = 20 m/s
   • Launch angle ($\theta$) = 45°
2. Convert the angle to radians: $$45°\times \frac{\pi }{180°}=0.7854\text{ radians}$$
3. Apply the range formula: $$R=\frac{(20\text{ m/s}{)}^2\times \sin (2\times 0.7854)}{9.81{\text{ m/s}}^2}$$
4. Simplify: $$R=\frac{400{\text{ m}}^2/{\text{s}}^2\times 1}{9.81{\text{ m/s}}^2}$$
5. Calculate: $$R=40.77\text{ m}$$

Example and Visual Representation

Let's visualize the projectile motion for the example above:

This diagram illustrates:

• The parabolic path of the projectile (red curve)
• The initial launch point (green circle)
• The launch angle of 45° (yellow dashed line)
• The initial velocity of 20 m/s
• The range of 40.77 meters (blue ground line)