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.

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)

Let's calculate the range for a projectile:

- Given:
- Initial velocity (\( v_0 \)) = 20 m/s
- Launch angle (\( \theta \)) = 45°

- Convert the angle to radians: \[ 45° \times \frac{\pi}{180°} = 0.7854 \text{ radians} \]
- Apply the range formula: \[ R = \frac{(20 \text{ m/s})^2 \times \sin(2 \times 0.7854)}{9.81 \text{ m/s}^2} \]
- Simplify: \[ R = \frac{400 \text{ m}^2/\text{s}^2 \times 1}{9.81 \text{ m/s}^2} \]
- Calculate: \[ R = 40.77 \text{ m} \]

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)

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