Projectile Motion Calculator
Range, maximum height, time of flight and impact velocity for a projectile at any speed, angle and launch height, with a plotted trajectory and the equations shown.
Projectile Motion Calculator: with the default inputs, horizontal range is 40.789 m.
Above horizontal is positive; negative means thrown downward.
Height of the launch point above the landing surface.
- Maximum height
- 10.197 mAbove the landing surface.
- Time of flight
- 2.884 s
- Impact speed
- 20 m/s
- Impact angle (below horizontal)
- 45 deg
- Time to peak
- 1.442 s
- Horizontal velocity
- 14.142 m/sConstant throughout the flight.
- Initial vertical velocity
- 14.142 m/s
Assumptions
- No air resistance, wind, spin or lift.
- Uniform gravity and a flat landing surface at height 0.
- Point mass — the object's size and rotation are ignored.
| Time(s) | Distance(m) | Height(m) | Vertical velocity(m/s) | Speed(m/s) |
|---|---|---|---|---|
| 0 s | 0 m | 0 m | 14.142 m/s | 20 m/s |
| 0.288 s | 4.079 m | 3.671 m | 11.314 m/s | 18.111 m/s |
| 0.577 s | 8.158 m | 6.526 m | 8.485 m/s | 16.492 m/s |
| 0.865 s | 12.237 m | 8.566 m | 5.657 m/s | 15.232 m/s |
| 1.154 s | 16.315 m | 9.789 m | 2.828 m/s | 14.422 m/s |
| 1.442 s | 20.394 m | 10.197 m | -0 m/s | 14.142 m/s |
| 1.731 s | 24.473 m | 9.789 m | -2.828 m/s | 14.422 m/s |
| 2.019 s | 28.552 m | 8.566 m | -5.657 m/s | 15.232 m/s |
| 2.307 s | 32.631 m | 6.526 m | -8.485 m/s | 16.492 m/s |
| 2.596 s | 36.71 m | 3.671 m | -11.314 m/s | 18.111 m/s |
| 2.884 s | 40.789 m | 0 m | -14.142 m/s | 20 m/s |
How this is worked out
The formula
vₓ = v₀ cos θ v_y = v₀ sin θ x(t) = vₓ t y(t) = h₀ + v_y t − ½ g t² Time of flight: t = [v_y + √(v_y² + 2 g h₀)] ÷ g Range: R = vₓ t Max height: H = h₀ + v_y² ÷ (2g) Impact speed: v = √(v₀² + 2 g h₀) Level ground (h₀ = 0): R = v₀² sin 2θ ÷ g, t = 2 v₀ sin θ ÷ g
Open How it’s calculated above to see this worked through with your own numbers.
What you enter
- Initial speed
- A number.in m/s · 0 or more · defaults to 20
- Launch angle
- Above horizontal is positive; negative means thrown downward.in deg, rad · from -90 to 90 · defaults to 45
- Initial height
- Height of the launch point above the landing surface.in m · 0 or more · defaults to 0
- Gravity(under More options)
- Earth 9.80665 (standard), Moon 1.62, Mars 3.71.from 0.01 to 1000 · defaults to 9.80665
What you get back
- Horizontal rangemain answer
- Maximum height
- Above the landing surface.
- Time of flight
- Impact speed
- Impact angle (below horizontal)
- Time to peak
- Horizontal velocity
- Constant throughout the flight.
- Initial vertical velocity
What this assumes
- No air resistance, wind, spin or lift.
- Uniform gravity and a flat landing surface at height 0.
- Point mass — the object's size and rotation are ignored.
About this calculator
Throw something and, ignoring air, its motion splits into two independent problems: constant velocity sideways and constant acceleration g downward. Everything else — how far it goes, how high, how long it is airborne, how hard it lands — follows from those two lines. Enter the launch speed, angle and height, and the calculator solves the kinematics exactly, plots the parabola, and lists position and velocity at ten points along the flight.
How to use it
Speed accepts m/s, km/h, mph, ft/s or knots; angle accepts degrees or radians; height is the launch point above where the object lands (a ball thrown from a 2 m hand, a shell fired from a cliff). Open More options to change gravity for the Moon or Mars. Use Solve for to work backwards — "what launch speed clears 40 m?" — on any output.
Reading the results
- Range is horizontal distance to the landing point at the landing height, not back at launch height.
- Maximum height is measured from the landing surface. A downward or level launch peaks at the launch point.
- Time to peak is v_y ÷ g; with no initial height the descent takes exactly as long as the ascent.
- Impact speed on level ground equals the launch speed — energy is conserved — and the impact angle mirrors the launch angle. From a height, it is larger and steeper.
Intuition worth keeping
On level ground 45° gives the longest range, and complementary angles (30° and 60°) land in the same place — one high and slow, one flat and fast. From an elevated launch, the optimum drops below 45°, because a flatter throw spends its time using the free height. The horizontal velocity never changes; the vertical velocity passes through zero at the peak, which is why the peak is the slowest point of the flight.
Where the model breaks
This is vacuum ballistics. Air drag grows with roughly the square of speed, so it hardly matters for a lobbed tennis ball but cuts a golf drive or a rifle bullet's range dramatically and makes the descent steeper than the climb. Spin (Magnus lift), wind and the Earth's curvature are also ignored. Treat the results as an upper bound for fast, light objects and as very accurate for slow, dense ones.
Frequently asked questions
▸What launch angle gives the maximum range?
45° on level ground with no air resistance, because sin 2θ peaks at θ = 45°. Launching from a height, the optimum is a little lower; with air drag, real optimums for balls are typically 35–42°.
▸How do I calculate time of flight?
Solve the vertical equation h₀ + v_y t − ½gt² = 0 for t. On level ground that reduces to t = 2v₀ sin θ ÷ g; from a height, use the quadratic formula — the calculator shows the substituted expression in the steps.
▸Why does the horizontal velocity stay constant?
With no air resistance nothing pushes horizontally, so by Newton's first law the horizontal component never changes. Gravity acts only on the vertical component.
▸Does the projectile's mass matter?
Not in vacuum — all masses fall with the same acceleration g, so the trajectory depends only on launch speed, angle and height. Mass matters once air resistance is included, which is why a feather and a stone diverge.
▸Is the impact speed the same as the launch speed?
Only when launch and landing are at the same height. From an elevated launch the object lands faster: v = √(v₀² + 2gh₀), regardless of angle.
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