Projectile Calculator
Range, peak height, and flight time from launch speed and angle.
What it does
Projectile motion splits launch velocity into horizontal coasting and vertical battling gravity. Range, peak, and hang time all follow from speed and angle through three closed formulas with no simulation needed.
The tool above takes launch speed with angle in meters or feet. It returns range, peak height, and flight time together for level ground throws.
The default of 20 m/s at 45° flies 40.79 m, peaks at 10.2 m, and lands after 2.88 s. The imperial mirror of 65.6 ft/s reads 133.8 ft with the same 2.88 s clock.
Equations follow NASA projectile references with the same derivation in Khan Academy physics units. Both assume vacuum and level ground; air drag and height differences need fuller models.
How to use
- Pick metric or imperial for your numbers.
- Enter the launch speed.
- Enter the angle above horizontal, 1 to 89.
- Read range first for the landing spot.
- Use height and time for clearance and timing.
Formula
- v² sin(2θ)
- Speed squared weighted by angle doubling.
- v sinθ
- Vertical launch component fighting gravity.
- g
- 9.80665 m/s² standard gravity.
Feet convert to meters first, then run metric physics, so systems agree. Angles near 0 or 90 collapse range or height as the math predicts.
Worked example
Example: 20 m/s at 45°
Metric defaults: 20 m/s, 45 degrees.
- Range: 400 x sin(90°) / 9.80665 = 400 / 9.80665 = 40.79 m.
- Vertical: 20 x sin(45°) = 14.14 m/s.
- Peak: 14.14² / 19.6133 = 200 / 19.6133 = 10.2 m.
- Time: 2 x 14.14 / 9.80665 = 2.88 s.
- Tool shows 40.79 m, 10.2 m, 2.88 s.
Reference table
| Angle | Range | Peak |
|---|---|---|
| 30° | 35.32 m | 5.10 m |
| 45° | 40.79 m | 10.20 m |
| 60° | 35.32 m | 15.3 m |
| 75° | 20.39 m | 19.03 m |
Limits
- Vacuum only; drag shortens and skews real arcs.
- Level landing assumed; raised targets land shorter.
- Constant g over the flight.
- Spin and lift (Magnus) sit outside.
Common mistakes
- Entering radians into degree math.
- Expecting 45° best with strong drag present.
- Forgetting landing height differs from launch.
- Mixing ft/s with metric g.
- Reading peak as half the range.
Try it yourself
Quick checks. Answers sit tucked under each question.
10 m/s at 30°?
Range 8.83 m, peak 1.27 m, time 1.02 s.
Why do 30° and 60° match range?
sin(2θ) ties them: sin 60° equals sin 120°.
Best vacuum angle?
45°, where sin(2θ) peaks at 1.
Questions
How far does 20 m/s at 45° fly?
40.79 m with a 10.2 m peak over 2.88 s in vacuum.
Why 45 degrees?
sin(2θ) maxes at 90°, so θ 45° maximizes vacuum range.
What changes with drag?
Shorter, skewed arcs favoring lower angles around 30 to 40°.
How is peak found?
Vertical speed squared over 2g: (v sinθ)² / 2g.
Height vs range?
Different curves: 60° outpeaks 45° while matching 30° range.
Uneven ground?
Landing above or below launch shifts range; this page assumes level.