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Projectile on an Incline Calculator

Projectile on an Incline Calculator

Projectile on an Incline Calculator

Calculate key parameters for a projectile launched on an incline (with the projectile moving upward relative to the 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: \[ R = \frac{2\,v_0^2\,\cos\theta\,\sin\left(\theta-\phi\right)}{g\,\cos^2\phi} \]
Maximum height above the incline: \[ h = \frac{v_0^2\,\sin^2\left(\theta-\phi\right)}{2\,g\,\cos\phi} \]
*Ensure that the launch angle (θ) is greater than the incline angle (φ).

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

Step 1: Enter Parameters

Example: 20 m/s

Example: 45° (measured from horizontal)

Example: 15° (angle of the incline relative to horizontal)

Example: 9.81 m/s²

Formulas:
Time of Flight: \( t = \frac{2\,v_0\,\sin(\theta-\phi)}{g\,\cos\phi} \)
Range along the Incline: \( R = \frac{2\,v_0^2\,\cos\theta\,\sin(\theta-\phi)}{g\,\cos^2\phi} \)
Maximum Height above the Incline: \( h = \frac{v_0^2\,\sin^2(\theta-\phi)}{2\,g\,\cos\phi} \)


Practical Example:
For an initial speed of 20 m/s, a launch angle of 45° (from horizontal), an incline of 15°, and \( g = 9.81 \) m/s²:
– Time of Flight ≈ \( \frac{2 \times 20 \times \sin(45°-15°)}{9.81 \times \cos15°} \approx 2.93\, \text{s} \)
– Range along the Incline ≈ \( \frac{2 \times 20^2 \times \cos45° \times \sin(30°)}{9.81 \times \cos^2(15°)} \approx 34.7\, \text{m} \)
– Maximum Height above the Incline ≈ \( \frac{20^2 \times \sin^2(30°)}{2 \times 9.81 \times \cos15°} \approx 3.5\, \text{m} \)

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