Projectile on an Incline (Downward Launch) Calculator

    Projectile on an Incline Calculator
    (Downward Launch)

    Calculate the parameters of a projectile launched from an elevated point above an inclined plane with a downward initial direction.
    The projectile is launched with an initial velocity \( v_0 \) at an angle \( \theta \) relative to the horizontal, the incline is at an angle \( \beta \), and the launch point is \( h \) meters above the plane.

    * Ensure that \( v_0\bigl(\sin\theta-\cos\theta\,\tan\beta\bigr) < 0 \) so that the projectile moves downward relative to the incline.

    Step 1: Enter Parameters

    Example: 20 m/s

    Angle relative to horizontal (should be small for downward launch)

    Angle of the incline relative to horizontal

    Height of the launch point above the plane

    Default: 9.81 m/s²

    Derived Formulas for Downward Launch:
    Intersection Equation: \( \frac{1}{2}gt^2 – v_0\Bigl(\sin\theta-\cos\theta\,\tan\beta\Bigr)t – h = 0 \)
    Time of Flight: \( T = \frac{v_0\Bigl(\sin\theta-\cos\theta\,\tan\beta\Bigr) + \sqrt{v_0^2\Bigl(\sin\theta-\cos\theta\,\tan\beta\Bigr)^2+2gh}}{g} \)
    Range along the Incline: \( R = x\cos\beta+y\sin\beta \) with \( x=v_0\cos\theta\,T \) and \( y=h+v_0\sin\theta\,T-\frac{1}{2}gT^2 \)


    Example:
    For \( v_0 = 20\,\text{m/s} \), \( \theta = 10^\circ \), \( \beta = 30^\circ \), \( h = 5\,\text{m} \), and \( g = 9.81\,\text{m/s}^2 \), the calculator computes the time of flight, range along the incline, impact speed, and shows the maximum height (which in a downward launch is the initial height).

     
    Projectile on an Incline Calculator
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