How to Calculate Projectile Motion

Projectile motion is the motion of an object thrown into the air, subject to only the acceleration of gravity. The key calculations include:

1. Horizontal and Vertical Velocities

- **Horizontal Velocity (v_x)**: \( v_x = v_0 \cos(\theta) \), where \( v_0 \) is the initial velocity and \( \theta \) is the launch angle.
- **Vertical Velocity (v_y)**: \( v_y = v_0 \sin(\theta) \)

2. Time of Flight

The total time the projectile is in the air is calculated by solving for when the vertical position reaches the ground.
\( t_{\text{flight}} = \frac{v_y + \sqrt{v_y^2 + 2g h}}{g} \)

3. Maximum Height

The maximum height is reached when the vertical velocity becomes zero. The maximum height can be calculated using: \( h_{\text{max}} = h + \frac{v_y^2}{2g} \)

4. Horizontal Range

The range is how far the projectile travels horizontally during its flight. It is calculated as: \( R = v_x \times t_{\text{flight}} \)

5. Visualizing the Trajectory

The trajectory of the projectile can be plotted using the equations of motion. At any time \( t \): - **Horizontal position**: \( x(t) = v_x \times t \) - **Vertical position**: \( y(t) = h + v_y \times t - 0.5 \times g \times t^2 \)