Projectile Motion Calculator

Projectile Motion Calculator solves for a missing variable in 2D projectile motion (no air resistance) using initial velocity, launch angle, initial height, gravity, and one of time, range, or maximum height.

Enter known values (leave the unknown field blank):

Initial Velocity \\( v_0 \\) (m/s, e.g., 20):

Launch Angle \\( \theta \\) (degrees, e.g., 45):

Initial Height \\( h_0 \\) (m, e.g., 0):

Gravitational Acceleration \\( g \\) (m/s², e.g., 9.81):

Time of Flight \\( t \\) (s, e.g., 4):

Horizontal Range \\( \Delta x \\) (m, e.g., 40):

Maximum Height \\( h_{\text{max}} \\) (m, e.g., 10):

Methodology Used in Projectile Motion Calculator

The calculator uses projectile motion equations (acceleration = \\(-g\\) in y-direction, zero in x-direction):

1. Horizontal: \\( \Delta x = v_0 \cos\theta \cdot t \\)

2. Vertical: \\( h = h_0 + v_0 \sin\theta \cdot t – \frac{1}{2} g t^2 \\)

3. Vertical velocity: \\( v_y = v_0 \sin\theta – g t \\)

4. Maximum height (\\( v_y = 0 \\)): \\( h_{\text{max}} = h_0 + \frac{(v_0 \sin\theta)^2}{2 g} \\)

5. Time to max height: \\( t_{\text{peak}} = \frac{v_0 \sin\theta}{g} \\)

Where:

\\( v_0 \\): Initial velocity (m/s)
\\( \theta \\): Launch angle (radians)
\\( h_0 \\): Initial height (m)
\\( h_{\text{max}} \\): Maximum height (m)
\\( \Delta x \\): Horizontal range (m)
\\( g \\): Gravitational acceleration (m/s²)
\\( t \\): Time of flight (s)

The solver identifies the missing variable, selects the appropriate equation, and solves, showing each step.

Example Calculation

Sample Input

Initial Velocity = 20 m/s, Launch Angle = 45°, Initial Height = 0 m, Gravity = 9.81 m/s², Time = ?

Step 1: Identify knowns: \\( v_0 = 20 \\), \\( \theta = 45^\circ \\), \\( h_0 = 0 \\), \\( g = 9.81 \\), solve for \\( t \\).

Step 2: Convert angle: \\( \theta = 45 \cdot \frac{\pi}{180} \approx 0.7854 \, \text{rad} \\)

Step 3: Use vertical equation at landing (\\( h = 0 \\)):

\\[ 0 = 0 + 20 \sin(0.7854) \cdot t – \frac{1}{2} \cdot 9.81 \cdot t^2 \\] \\[ 4.905 t^2 = 20 \cdot 0.7071 \cdot t \approx 14.142 t \\] \\[ t (4.905 t – 14.142) = 0 \\] \\[ t = 0 \, \text{or} \, t = \frac{14.142}{4.905} \approx 2.88 \, \text{s} \\]

Result: Time of Flight = 2.88 s.