Projectile Motion Calculator
Calculate the range, maximum height, and flight time of a projectile given its initial velocity, launch angle, and initial height.
About this calculator
This calculator solves classic 2D projectile motion under constant gravity, letting the launch point sit above ground level rather than assuming a flat launch. It splits initial velocity into horizontal (vx = v·cos θ) and vertical (vy = v·sin θ) components, then solves the vertical position equation 0 = h₀ + vy·t − ½gt² for the positive root using the quadratic formula — this is what lets a projectile launched from a height come down further than one launched from the ground at the same speed and angle. Range is simply the horizontal component times that flight time, since horizontal velocity never changes without air resistance. Maximum height adds the extra rise (vy²/2g) on top of the starting height, and time-to-apex (vy/g) is the point where vertical velocity crosses zero.
Impact speed recombines the unchanged horizontal velocity with the vertical velocity computed from the same energy relationship used for flight time, giving the speed at which the projectile strikes the ground. The trajectory chart samples 50 points across the flight to plot the parabolic path. Key assumption: no air resistance, so the classic 45° angle only maximizes range on flat, level ground — the moment initial height is nonzero, or if drag needs to be accounted for, the optimal angle shifts and this simplified model no longer captures it exactly. It also assumes the landing point is at the same flat elevation as the ground reference, rather than a sloped or elevated target.
Inputs
Results
Horizontal Range
40.77 m
≈ 9 car lengths
Maximum Height
10.19 m
≈ 6 adult heights
Total Flight Time
2.88 s
How to Use This Calculator
- Enter Initial Velocity, Launch Angle, and Initial Height.
- Review Horizontal Range (m), Maximum Height, and Total Flight Time (s).
- Use Time to Max Height (s) and Impact Speed (m/s) to inform your decision.
- Use the chart to visualize the results and explore different scenarios by adjusting inputs.
How the result changes with Initial Velocity
| Initial Velocity | Horizontal Range | Maximum Height | Total Flight Time |
|---|---|---|---|
| 10 | 10.19 m | 2.55 m | 1.44 s |
| 15 | 22.94 m | 5.73 m | 2.16 s |
| 30 | 91.74 m | 22.94 m | 4.33 s |
| 50 | 254.84 m | 63.71 m | 7.21 s |
What each input means
- Initial Velocity
- Speed at which the projectile is launched.
- Launch Angle
- Angle above the horizontal in degrees. 45° gives maximum range on flat ground.
- Initial Height
- Height above ground level at launch. Set to 0 for ground-level launches.
How this is calculated
Worked example, using the default values
- Identify Input ParametersInitial Velocity = 20, Launch Angle = 45, Initial Height = 0 = 3 input(s) provided
- Calculate Horizontal RangeHorizontal Range40.77 = 40.77
- Calculate Maximum HeightMaximum Height10.19 = 10.19
- Calculate Total Flight TimeTotal Flight Time2.883 = 2.883
- Calculate Time to Max HeightTime to Max Height1.442 = 1.442
- Calculate Impact SpeedImpact Speed20 = 20
Engine last updated . Built by Paul Gunder, a software engineer, not a licensed financial, medical, or legal professional.
Frequently Asked Questions
Why is impact speed higher than initial velocity when I launch from a height?
Horizontal velocity never changes during the flight, but the vertical velocity component keeps growing as gravity accelerates the projectile downward, and it's larger when falling from an added starting height than it would be launching from ground level at the same speed. Impact speed combines that larger vertical component with the unchanged horizontal one, so launching from any height above zero always produces an impact speed greater than the initial launch speed.
Is 45° always the best angle for maximum range?
Only when initial height is zero — the calculator's own model shows that with a nonzero launch height, the extra fall time before impact favors a slightly flatter angle than 45°, since more of the initial speed spent going horizontal pays off before gravity has as much time to act. This calculator will compute range correctly at any angle you choose, but it doesn't automatically find the true optimal angle for a nonzero height for you.
Why does Time to Max Height not depend on the initial height I entered?
Time to apex is computed as vy/g — the point where the vertical velocity component crosses zero — which depends only on initial vertical velocity and gravity, not on where that velocity started from. A projectile launched from a tall platform reaches its peak vertical speed loss at the same elapsed time as one launched from the ground with the same speed and angle; it's just starting (and ending) higher up.
Does this calculator account for air resistance or wind?
No — it's a pure constant-gravity, no-drag model, so horizontal velocity is assumed to never change during flight. Real projectiles with significant air resistance (a baseball, a bullet, anything not extremely dense and streamlined) will have a shorter range and steeper descent than this model predicts, and the optimal launch angle for maximum range in reality is typically lower than 45°.
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