Projectile Range Calculator
Enter the launch velocity and angle - or any two known quantities - to instantly find how far the projectile travels, how high it rises, how long it is in the air, and the velocity at landing. Switch between metric and imperial units, try different launch angles, and use the trajectory chart to see the full flight path.
Projectile motion - the physics
A projectile is any object launched into the air and then left to move under gravity alone (no thrust, no air resistance in the standard model). The motion splits cleanly into two independent parts: horizontal motion at constant velocity, and vertical motion under constant gravitational acceleration. Because the two directions are independent, you can solve each separately and combine the results. The horizontal range depends on three things: the launch speed, the launch angle, and how long the projectile stays airborne. A higher launch point extends flight time and therefore increases range even without changing the launch speed.
The range formula explained
For a launch from ground level (h = 0), the horizontal range simplifies to R = V0^2 * sin(2a) / g, where V0 is the launch speed, a is the launch angle above horizontal, and g is gravitational acceleration. This formula shows why 45 degrees gives the maximum range - sin(2 * 45) = sin(90) = 1, the largest possible value. It also explains the complementary-angle symmetry: sin(2a) = sin(180 - 2a), so 30 and 60 degrees produce identical ranges on flat ground. When the launch height is above zero, the full formula is R = Vx * [Vy + sqrt(Vy^2 + 2*g*h)] / g, where Vx and Vy are the horizontal and vertical velocity components and h is the launch height.
How to use the different solve modes
Velocity + Angle is the standard mode: enter what you know about the throw or shot and read off the range, height, and time. Horizontal + Vertical velocity suits cases where you have measured the two components separately, for example from video analysis or a radar gun with angle data. Velocity + Range back-solves for the launch angle that hits a specific target distance at the given speed - useful for aiming problems. Angle + Range finds the minimum speed needed to reach a target at the given angle, which comes up in robotics, sports science, and military applications. All four modes share the same underlying physics; the calculator simply rearranges which variable is the unknown.
Gravity, planets, and real-world corrections
The gravity selector lets you explore how the same throw plays out on the Moon (1.62 m/s^2), Mars (3.72 m/s^2), or Jupiter (24.79 m/s^2). Because range is inversely proportional to g, a ball thrown at 30 m/s at 45 degrees travels about 91.7 m on Earth but roughly 553 m on the Moon. In the real world, air resistance always reduces range below the vacuum model prediction. The drag effect is most significant for light objects with large cross-sectional areas (a badminton shuttlecock, a foam ball) and least significant for dense, compact objects (a golf ball, a cannon ball) at moderate speeds.
Projectile range by launch angle (V0 = 30 m/s, level ground)
| Angle (deg) | Range (m) | Max height (m) | Time of flight (s) |
|---|---|---|---|
| 10 | 31.2 | 4.5 | 1.06 |
| 20 | 58.9 | 17.4 | 2.10 |
| 30 | 79.5 | 34.4 | 3.06 |
| 45 | 91.7 | 57.3 | 4.33 |
| 60 | 79.5 | 102.9 | 5.30 |
| 70 | 58.9 | 120.8 | 5.74 |
| 80 | 31.2 | 136.3 | 5.99 |
How the horizontal range changes with launch angle at a fixed launch speed of 30 m/s on Earth. Maximum range occurs at 45 degrees.
Frequently asked questions
What angle gives the maximum projectile range?
45 degrees above horizontal gives the maximum range when launching from and landing on the same level with no air resistance. This is because the range formula includes the factor sin(2a), which reaches its maximum value of 1 at 2a = 90 degrees, so a = 45 degrees. When the launch point is elevated above the landing surface, the optimal angle is slightly less than 45 degrees because the extra height already gives the projectile additional flight time.
Why do two different angles give the same range?
The range formula on level ground is R = V0^2 * sin(2a) / g. Since sin(x) = sin(180 - x), any pair of angles that add up to 90 degrees will produce the same range. For example, 30 and 60 degrees give the same horizontal distance. The difference is that the steeper angle results in a much higher peak altitude and longer time in the air, while the shallower angle keeps the trajectory flatter and the flight time shorter.
Does the mass of the projectile affect the range?
In the standard vacuum model, no - mass cancels out of the equations entirely, so a feather and a cannonball with the same launch conditions would travel the same distance. In reality, air resistance makes a large difference because lighter or less aerodynamic objects decelerate faster. The vacuum model is a good approximation for dense, compact objects at moderate speeds but can significantly overestimate range for light or wide objects.
How do I convert horizontal and vertical velocity into speed and angle?
If you know the horizontal component Vx and the vertical component Vy at launch, the total launch speed is sqrt(Vx^2 + Vy^2) and the launch angle is arctan(Vy / Vx) in degrees. This calculator does that conversion automatically when you choose the "Horizontal + Vertical velocity" mode.
How does launch height affect the range?
A higher launch point extends the time the projectile is in the air, which increases the horizontal distance it covers even at the same speed and angle. The full formula is R = Vx * [Vy + sqrt(Vy^2 + 2*g*h)] / g, where h is the height above the landing surface. This means that for targets below the launch point, you also get more range, which is why artillery on a hillside or a ball thrown off a cliff travels further than the same throw from ground level.