Trajectory Calculator - Projectile Motion
Enter your launch speed, angle, and initial height to instantly get the horizontal range, maximum height, total flight time, and both velocity components for ideal projectile motion. Switch between metric and imperial units at any time.
What is projectile motion?
Projectile motion describes the path of any object launched into the air under the influence of gravity alone. Once released, horizontal velocity remains constant (no horizontal forces), while vertical velocity changes steadily due to gravitational acceleration. The combination produces a parabolic trajectory. Real-world scenarios include a ball thrown at an angle, a bullet fired from a gun, a water jet from a hose, and a ski jumper leaving a ramp. This calculator assumes ideal conditions: constant gravity, flat ground at the landing point, and no air resistance.
How to use this calculator
Enter the initial velocity (the speed at the moment of launch), the launch angle above the horizontal, and the initial height if the projectile is launched from an elevated position. Choose metric (m/s, metres) or imperial (ft/s, feet). You can also change the gravity preset to compare what the same throw would achieve on the Moon or on Mars. The results update instantly: horizontal range, maximum height above ground, total flight time, and the two velocity components. The "Show your work" panel below the results walks through every formula step by step with your actual numbers.
Projectile motion formulas
The horizontal and vertical positions at time t are: x(t) = V0 cos(alpha) * t and y(t) = h0 + V0 sin(alpha) * t - 0.5 * g * t^2. The flight time T is found by solving y(T) = 0, giving T = [V0 sin(alpha) + sqrt((V0 sin(alpha))^2 + 2 g h0)] / g. The maximum height is h_max = h0 + (V0 sin(alpha))^2 / (2g). The horizontal range is R = V0 cos(alpha) * T. For a flat launch from ground level (h0 = 0), these simplify to T = 2 V0 sin(alpha) / g and R = V0^2 sin(2 alpha) / g.
Complementary angles and the 45-degree optimum
A remarkable property of projectile motion (with h0 = 0) is that two launch angles symmetric around 45 degrees - such as 30 and 60 degrees, or 20 and 70 degrees - produce exactly the same range. The 45-degree angle gives the longest range because it balances horizontal speed and hang time optimally. However, when the launch point is above the landing point the optimal angle shifts below 45 degrees, and when launching upward to a higher target it shifts above 45 degrees. The gravity preset lets you explore how lower gravity (Moon, Mars) stretches every distance proportionally.
Launch angle vs. range (relative, same V0)
| Launch angle | Relative range | Notes |
|---|---|---|
| 0 degrees | 0% | Horizontal - zero height gain |
| 15 degrees | 50% | Complementary to 75 degrees |
| 30 degrees | 87% | Complementary to 60 degrees |
| 45 degrees | 100% | Maximum range |
| 60 degrees | 87% | Complementary to 30 degrees |
| 75 degrees | 50% | Complementary to 15 degrees |
| 90 degrees | 0% | Straight up - falls back to start |
Range as a fraction of maximum range (at 45 degrees), for a flat surface and no air resistance.
Frequently asked questions
Why does 45 degrees give the maximum range?
For a projectile launched from and landing at the same height, the range formula is R = V0^2 * sin(2*alpha) / g. The sine function reaches its maximum of 1 when its argument is 90 degrees, meaning 2*alpha = 90 degrees, so alpha = 45 degrees. Any angle higher or lower reduces sin(2*alpha) and therefore reduces the range.
What happens when the launch height is not zero?
When the projectile starts above the ground (h0 > 0), the extra height gives it more time in the air. The flight-time formula becomes a quadratic equation, and its solution involves a square root term that depends on h0. Longer time in the air means more horizontal distance covered, so the optimal angle shifts below 45 degrees. This calculator handles any initial height automatically.
Does air resistance matter?
In real life, air resistance (drag) reduces both range and maximum height significantly, especially at high speeds. The effect scales with the square of speed and depends on the object's shape and size. This calculator uses the ideal, drag-free model that is standard for physics courses and good for small, slow objects. For precision engineering or ballistics, a drag coefficient must be included.
How does gravity on the Moon compare to Earth?
The Moon's surface gravity is about 1.62 m/s^2, roughly 1/6 of Earth's 9.807 m/s^2. Because range and max height scale inversely with g, the same launch on the Moon produces about 6 times the range and 6 times the maximum height. The Moon preset lets you see this directly.
Can I find the launch angle if I know the range I want?
For a flat-ground launch, the range formula R = V0^2 * sin(2*alpha) / g can be rearranged to alpha = 0.5 * arcsin(R * g / V0^2). There are always two solutions (symmetric around 45 degrees) unless R equals the maximum. Enter different angles and observe the range output to find the angle that meets your target.