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.

    Kinematics Tutorial: Using a Parabolic Projectile from an Elevation Calculator

    Real‑world projectiles (baseballs, golf shots, artillery shells) are often launched from a height above (or below) the landing surface. Adding the initial elevation \(y_0\) changes flight time, range, and maximum height compared with “flat‑ground” textbook cases. The Parabolic Projectile from an Elevation Calculator solves the kinematic equations instantly—displaying time of flight, horizontal range, peak height, and trajectory plots. This tutorial reviews the underlying physics and shows you how to use the online tool at freeonlinecalculators.net.

    Why Does Launch Elevation Matter?

    • Longer flight time: Starting above ground means extra “fall distance,” so the projectile stays airborne longer.
    • Greater range (for upward elevation): More airtime → larger horizontal distance, even with the same launch speed and angle.
    • Safety and engineering: Designing ballistics, ski‑jump ramps, or drone deliveries requires accounting for non‑zero launch/landing heights.

    Key Equations (with initial elevation \(y_0\))

    The projectile’s position vs time is

    \( x(t) = v_0\cos\theta\,t,\qquad y(t) = y_0 + v_0\sin\theta\,t – \tfrac{1}{2}gt^2 \)

    Time of Flight \(t_\text f\)

    Solve \(y(t_\text f)=0\) (ground level):

    \( t_\text f = \frac{v_0\sin\theta + \sqrt{(v_0\sin\theta)^2 + 2g\,y_0}} {\,g\,} \)

    Horizontal Range \(R\)

    \( R = v_0\cos\theta \; t_\text f \)

    Maximum Height \(y_\text{max}\)

    Occurs at \(t_\text{peak}=v_0\sin\theta/g\):

    \( y_\text{max} = y_0 + \frac{(v_0\sin\theta)^2}{2g} \)

    Constants: \(g = 9.81\;\text{m s}^{-2}\) (default; user can change).

    How to Use the Online Parabolic Projectile from Elevation Calculator

    1. Open the calculator here.
    2. Enter:
      • Initial Speed (v₀)
      • Launch Angle (θ) (degrees or radians)
      • Launch Elevation (y₀) (positive = above ground, negative = below)
      • Gravity (g) (optional; default 9.81 m/s²)
    3. Click Calculate.
    4. Read time of flight, range, peak height, and a trajectory plot.
    5. Use Reset to start a new scenario.

    Example Problems

    Example 1 — Golf Shot from a 10 m Cliff

    Given: \(v_0 = 20\;\text{m/s}, θ=45°, y_0 = 10\;\text m\).
    \(v_0\sinθ = 14.14\), \(v_0\cosθ = 14.14\).
    \(t_\text f = \frac{14.14 + \sqrt{14.14^2 + 2(9.81)(10)}}{9.81} \approx 3.47\;\text s\).
    \(R = 14.14 \times 3.47 \approx 49.1\;\text m\).
    \(y_\text{max} = 10 + \frac{14.14^2}{2(9.81)} \approx 20.2\;\text m\).

    Example 2 — Soccer Kick from Ground Level

    \(v_0 = 28\;\text{m/s}, θ=30°, y_0 = 0\).
    \(t_\text f = \frac{v_0\sinθ}{g} \times 2 = 2v_0\sinθ/g \approx 2.86\;\text s\).
    \(R = v_0\cosθ \, t_\text f \approx 69.4\;\text m\).
    \(y_\text{max} = \frac{(v_0\sinθ)^2}{2g} \approx 10.0\;\text m\).

    Frequently Asked Questions

    Does the calculator include air resistance?

    No—results assume ideal projectile motion (no drag, no wind). At typical sports speeds the error is small; at high velocities or long ranges, air resistance must be modeled separately.

    What happens if \(y_0\) is negative?

    A negative launch elevation means the projectile starts below the landing plane (e.g., throwing a ball onto a ledge). The same formulas apply—time of flight decreases.

    How do I find the angle for maximum range?

    With \(y_0 \neq 0\), the optimal angle is below 45° (above ground) or above 45° (below ground). The calculator can sweep angles to show range vs angle, or you can set Optimize Angle to let it pick the maximum automatically.

    Can I change gravity for the Moon or Mars?

    Yes—override g (1.62 m/s² for the Moon, 3.71 m/s² for Mars) to simulate extraterrestrial trajectories.

     
    Parabolic Projectile Calculator( launched on level ground)
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