Parabolic Projectile from an Elevation Calculator

Parabolic Projectile from an Elevation Calculator

Calculate the parameters of a projectile launched from an elevated position.
Enter the initial velocity \( v_0 \), launch angle \( \theta \), initial elevation \( h \), and gravitational acceleration \( g \).

* Ensure all values are in SI units: m/s for velocity, m for elevation, and m/s² for gravity.

Step 1: Enter Parameters

Example: 20 m/s

Angle relative to horizontal

Height above ground level

Default: 9.81 m/s²

Derived Formulas:
Time of Flight: \( T = \frac{v_0\sin\theta+\sqrt{(v_0\sin\theta)^2+2gh}}{g} \)
Maximum Height: \( H_{\text{max}} = \begin{cases} h+\frac{(v_0\sin\theta)^2}{2g} & \text{if } v_0\sin\theta>0 \\ h & \text{if } v_0\sin\theta\le0 \end{cases} \)
Range: \( R = v_0\cos\theta\,T \)
Impact Speed: \( v_{\text{impact}} = \sqrt{(v_0\cos\theta)^2+\left(v_0\sin\theta-gT\right)^2} \)


Example:
For \( v_0 = 20\,\text{m/s} \), \( \theta = 45^\circ \), \( h = 10\,\text{m} \), and \( g = 9.81\,\text{m/s}^2 \), the calculator computes the time of flight, maximum height, range, and impact speed.