Projectile Motion Calculator
Enter launch speed, angle and height to get range, maximum height, time of flight and impact speed, with the trajectory plotted and angles to hit a target.
The interactive tool loads here when JavaScript runs.
About this toolWhat is the Projectile Motion Calculator?
This projectile motion calculator takes a launch speed, a launch angle and a launch height and returns where the projectile lands, how high it climbs, how long it is in the air and how fast and steeply it comes down. The trajectory is drawn as you type, with the top of the arc marked.
It also answers the two questions that come up most after the basic numbers. Which angle throws farthest from this height? (45° only on level ground; from a raised launch point it is less.) And which angles land on a target at a given distance? There are usually two, a flat direct shot and a high lob, and the calculator gives both or tells you the target is out of reach.
Speeds can be in m/s, km/h, mph, ft/s or knots and distances in metres, centimetres, kilometres, feet or yards. Gravity defaults to standard Earth gravity and can be switched to the Moon, Mars, other planets or a custom value.
- Range, peak height, flight time, impact speed and angle
- Launch from any height, angles from −90° to 90°
- Best angle for maximum range from that height
- Angles to hit a target at a given distance and height
- Plotted trajectory, planet gravity presets, copy and share
How to use the Projectile Motion Calculator
- Enter the launch speed and choose its unit.
- Enter the launch angle above horizontal in degrees. Use a negative angle for a downward throw and 0 for a horizontal launch.
- Enter the launch height above the landing level (0 for level ground) and pick the distance unit, which applies to every length on the page.
- Read the range, maximum height, flight time, impact speed and angle, and the best angle for distance, then check the plotted path.
- Under Hit a target, enter the target distance and height to get the low and high launch angles that land on it.
Worked examples
Level ground at 45°
Input
20 m/s, 45°, height 0
Output
Range 40.79 m, max height 10.20 m, time 2.884 s
R = v²/g = 400/9.80665. Maximum height is a quarter of the range at 45°.
Horizontal launch from a ledge
Input
10 m/s, 0°, height 20 m
Output
Lands 20.20 m out after 2.020 s at 22.19 m/s, 63.2° below horizontal
The fall time is the same as dropping from 20 m. A 24.3° launch would reach 22.62 m.
Shot put
Input
13.7 m/s, 37°, release height 2.1 m
Output
Range 20.86 m; best angle 42.2° for 21.13 m
In practice athletes release lower than the drag-free optimum because they can push harder at flatter angles.
Two ways to hit a target
Input
20 m/s from 1.5 m, target 30 m away on the ground
Output
20.2° or 67.0°
The low angle gets there sooner; the lob takes far longer and comes down steeply.
Background and insight
Without air resistance, a projectile's horizontal and vertical motions are independent. The horizontal velocity v cos θ never changes, and the vertical velocity v sin θ falls by g every second, exactly as Galileo described in his Two New Sciences (1638). Combining the two gives a parabola, y = h + x tan θ − g x² ÷ (2 v² cos² θ).
Time of flight comes from solving the vertical equation for the moment the projectile returns to the landing level: t = (v sin θ + √(v² sin² θ + 2 g h)) ÷ g. On level ground (h = 0) this reduces to 2 v sin θ ÷ g, the range becomes v² sin 2θ ÷ g, and the maximum range is v² ÷ g at 45°. Complementary angles such as 30° and 60° then land at the same spot.
From a height h, the best angle drops below 45° and is given by tan θ = v ÷ √(v² + 2 g h), with a maximum range of (v ÷ g) √(v² + 2 g h). Athletes know this intuitively: shot putters release well below 45°, typically in the mid to high 30s. Air drag lowers the best angle further and shortens every range, by a few percent for a heavy shot and by a third or more for a fast baseball or golf ball, so treat drag-free results as an upper bound.
Common use cases
Physics and engineering courses
Check kinematics answers for range, height and time, and see the parabola for each launch.
Sports analysis
Explore how release angle and height change the distance of a throw, kick or jump.
Water jets and fountains
Estimate how far a nozzle at a given exit speed and angle will reach.
Game and simulation design
Get launch velocities and angles that land a projectile exactly on a target.
Tips and limitations
- Results ignore air resistance and spin. They are close for slow, dense objects and optimistic for fast, light ones.
- Measure the launch height from the level where the projectile lands, not from the ground under the thrower if the landing area is lower or higher.
- If the target is above the launch point, enter the target height above the landing level; the calculator converts it to a height relative to launch.
- For a dropped object (speed 0), use the free fall calculator instead, which also models drag.
- Rounding g to 9.8 instead of 9.80665 changes the answers by about 0.07%, which matters only when comparing with a textbook answer key.
Frequently asked questions
What angle gives the maximum range?
On level ground with no air resistance, 45°. Launching from a height lowers the best angle: tan θ = v ÷ √(v² + 2gh). Air drag lowers it further, which is why real throws often peak between 35° and 42°.
How do you calculate the time of flight?
Solve the vertical motion for the landing time: t = (v sin θ + √((v sin θ)² + 2gh)) ÷ g. On level ground that is 2v sin θ ÷ g; 20 m/s at 45° stays up 2.884 s.
How do I find the maximum height of a projectile?
H = h + (v sin θ)² ÷ (2g). At 20 m/s and 45° from the ground, the vertical speed is 14.14 m/s, so H = 200 ÷ 19.61 = 10.20 m.
Why do two angles hit the same target?
Range on level ground depends on sin 2θ, which has the same value for θ and 90° − θ. So 30° and 60° land together. With a height difference the two angles are no longer complementary, but there are still usually two.
Does the mass of the projectile matter?
Not without air resistance: every mass follows the same path for the same launch speed and angle. With drag, heavier objects of the same size are slowed less and travel farther.
How accurate is this for a real ball?
It is an upper bound. A shot put lands within a few percent of the prediction, but a baseball hit at 45 m/s travels well under the 200 m the vacuum formula gives, because drag removes a large share of its speed.
Is there an in-depth guide to the Projectile Motion Calculator?
Yes. “Projectile Motion Equations: Range, Height, Time and the Best Angle” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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