Projectile Range Calculator
Find the horizontal range of a projectile, the launch angle that maximises it, or the launch speed needed to hit a target distance.
🎯 What is Projectile Range?
Projectile range is the horizontal distance a launched object travels before it returns to the landing level, when only gravity acts on it. Launch a ball, a stone, a javelin or a cannonball with speed v₀ at angle θ above the horizontal and it follows a parabola. The range is how far along the ground that parabola reaches. It is set by three things: how fast you launch, at what angle, and how strong gravity is. If the launch point is higher than the landing point, the height also matters, because the object keeps travelling sideways while it falls the extra distance.
Range questions come up in physics classes, sports analysis and engineering. A teacher asks how far a cannonball travels at a given angle. A coach wants to know the best release angle for a throw or a kick. A designer of a water fountain, a sprinkler or a launcher needs the speed that reaches a target some distance away. This calculator answers all of these: the first mode gives range, flight time, maximum height and impact conditions, and the second mode works backward from a target distance to the launch speed you need.
The most famous result is that 45 degrees gives the longest range on level ground, and that angles adding up to 90 degrees (for example 30 and 60) land at the same spot. Both facts come straight from the formula R = v₀² sin(2θ) / g, and the calculator shows them with your own numbers. A less known result is that the best angle falls below 45 degrees when the launch point is above the landing point. A shot putter releasing from shoulder height, or a stone thrown off a cliff, does best with a flatter throw. The calculator finds this optimum from the exact formula and also reports how close your chosen angle comes to the maximum.
All results assume an ideal projectile: no air resistance, a flat landing surface and constant gravity. That is the standard textbook model and it is accurate for dense objects at modest speed. Real balls, arrows and bullets lose range to drag, so treat the output as an upper bound. For the full trajectory with horizontal and vertical velocity components, see the projectile motion calculator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Cannonball in Physics Class
v₀ = 50 m/s at 45° from level ground on Earth
Example 2 — A Long Jump
v₀ = 9 m/s at 20°, treated as level ground on Earth
Example 3 — A Football Punt
v₀ = 25 m/s at 50° from a 0.5 m release height on Earth
Example 4 — A Javelin Released at 2 m
v₀ = 28 m/s at 35° from a 2 m release height on Earth
Example 5 — The Same Launch on the Moon
v₀ = 20 m/s at 45° on the Moon (g = 1.62 m/s²)
Example 6 — Speed Needed for a 60 m Throw
Target 60 m at 40° from a 2 m release height on Earth
Example 7 — Speed Needed for a 100 m Shot at 45°
Target 100 m at 45° from level ground on Earth
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for the range of a projectile?
On level ground, R = v0^2 sin(2 theta) / g. When launched from a height h0 above the landing level, R = (v0 cos theta / g) times (v0 sin theta + sqrt((v0 sin theta)^2 + 2 g h0)). Here v0 is launch speed, theta the launch angle and g the gravitational acceleration, with air resistance ignored.
What angle gives the maximum range?
On level ground the maximum range occurs at 45 degrees because sin(2 theta) is largest (equal to 1) when theta is 45 degrees. For a launch from height h0 the best angle is arctan(v0 / sqrt(v0^2 + 2 g h0)), which is always less than 45 degrees.
Why do 30 and 60 degrees give the same range?
On level ground the range depends on sin(2 theta), and sin(2 x 30) = sin(60) equals sin(2 x 60) = sin(120). Any pair of angles adding up to 90 degrees has the same sine of twice the angle, so they land at the same point. The lower angle gives a flatter, quicker flight and the higher angle a higher, slower one.
How does launch height change the optimum angle?
A raised launch point adds fall time, so the best angle moves below 45 degrees. For v0 = 28 m/s and a 2 m release height the optimum is about 44.3 degrees, and for a 50 m cliff with v0 = 20 m/s it is about 28.3 degrees. Level ground gives exactly 45 degrees.
How do you find the launch speed for a target range?
Set the vertical position to zero at the target and solve for v0. On level ground v0 = sqrt(R g / sin(2 theta)). With a launch height, v0^2 = g R^2 / (2 cos^2(theta) (h0 + R tan(theta))). The speed mode of this calculator does that and also reports the lowest possible speed over all angles.
What is the maximum range for a given speed?
On level ground the maximum range is v0^2 / g, reached at 45 degrees. From height h0 it is (v0 / g) times sqrt(v0^2 + 2 g h0). For v0 = 20 m/s on Earth with h0 = 0 that is about 40.8 m.
Does mass affect projectile range?
Not without air resistance. Gravity gives every mass the same acceleration, so a heavy and a light object launched with the same speed and angle follow the same path. In real air, drag acts differently on different objects, so light or large objects fall short of the ideal range.
How far does a projectile go on the Moon?
With g = 1.62 m/s2 instead of 9.80665, range is about 6 times larger for the same speed and angle (the ratio of the g values is 6.05). A 20 m/s launch at 45 degrees goes about 246.9 m on the Moon and 40.8 m on Earth.
Does air resistance change the best launch angle?
Yes. With drag the optimum angle is usually a little below 45 degrees on level ground and the range is shorter than the ideal formula says. This calculator uses the ideal vacuum model, which is a good guide for dense, slow objects such as a shot put or a cannonball at classroom speeds.
What is the impact angle and why does it matter?
The impact angle is the angle below horizontal at which the projectile arrives, found from the vertical and horizontal velocity components at landing. On level ground it equals the launch angle. From a raised launch point it is steeper than the launch angle.