Projectile on an Incline (Upward Motion) Calculator

Projectile on an Incline (Upward Motion) Calculator

Calculate the key parameters for a projectile launched upward on an inclined plane.
The equations used are:
Time of Flight: \[ t = \frac{2\,v_0\,\sin\left(\theta-\phi\right)}{g\,\cos\phi} \]
Range along the Incline: \[ R = \frac{2\,v_0^2\,\sin\left(\theta-\phi\right)\cos\theta}{g\,\cos^2\phi} \]
Maximum Height above the Incline: \[ H_{max} = \frac{v_0^2\,\sin^2\left(\theta-\phi\right)}{2\,g\,\cos\phi} \]

* Enter initial velocity (m/s), launch angle (°), incline angle (°), and gravitational acceleration (m/s²). Ensure that the launch angle is greater than the incline angle.

Step 1: Enter Parameters

Example: 20 m/s

Example: 60° (from horizontal)

Example: 30° (angle of the incline)

Example: 9.81 m/s²

Equations used:
Time of Flight: \( t = \frac{2\,v_0\,\sin(\theta-\phi)}{g\,\cos\phi} \)
Range: \( R = \frac{2\,v_0^2\,\sin(\theta-\phi)\cos\theta}{g\,\cos^2\phi} \)
Maximum Height: \( H_{max} = \frac{v_0^2\,\sin^2(\theta-\phi)}{2\,g\,\cos\phi} \)


Practical Example:
For an initial velocity of 20 m/s, a launch angle of 60°, an incline angle of 30°, and \( g=9.81 \) m/s²:
– Time of Flight ≈ \( \frac{2 \times 20 \times \sin(60°-30°)}{9.81 \times \cos30°} \approx \frac{40 \times \sin30°}{9.81 \times 0.866} \approx \frac{40 \times 0.5}{8.50} \approx 2.35\, \text{s} \)
– Range along the Incline ≈ \( \frac{2 \times 20^2 \times \sin30° \times \cos60°}{9.81 \times \cos^2 30°} \approx \frac{800 \times 0.5 \times 0.5}{9.81 \times 0.75} \approx 27.18\, \text{m} \)
– Maximum Height above the Incline ≈ \( \frac{20^2 \times \sin^2 30°}{2 \times 9.81 \times \cos30°} \approx \frac{400 \times 0.25}{19.62 \times 0.866} \approx 5.91\, \text{m} \)

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