Calculate the angle of projection, range, or initial velocity from any two values using m/s, km/h, mph, ft/s, meters, km, miles, yards, feet, degrees, or radians.

Level launch and landing height, no air resistance, g = 9.81 m/sยฒ. Both complementary launch angles are shown.

Related Calculators

Angle Of Projection Formula

The angle of projection calculator uses the standard projectile range equation for a projectile launched and landing at the same height, with air resistance ignored.

R = (vยฒ ร— sin(2 ร— theta)) / g
v = sqrt((R ร— g) / sin(2 ร— theta))
theta = arcsin((R ร— g) / vยฒ) / 2
  • R = horizontal range of the projectile
  • v = initial velocity
  • theta = angle of projection
  • g = acceleration due to gravity, usually 9.81 m/s2

Select Launch angles, Horizontal range or Initial speed, then enter the two visible values. If you enter initial velocity and angle, the calculator finds the range. If you enter range and angle, it rearranges the same equation to find the required initial velocity. If you enter initial velocity and range, it solves for the angle of projection.

For a given velocity and range, there can be two possible launch angles: a lower angle and a higher complementary angle. The calculator shows both the lower principal angle and the higher complementary angle. They coincide at 45ยฐ for the maximum range.

Common Projectile Angle Results

For level-ground projectile motion, the range depends on sin(2ฮธ). The largest range occurs at 45 degrees when air resistance is ignored.

Angle sin(2ฮธ) Range compared with maximum
15ยฐ 0.500 50%
30ยฐ 0.866 86.6%
45ยฐ 1.000 100%
60ยฐ 0.866 86.6%
75ยฐ 0.500 50%

Unit Conversions Used

Quantity Unit Base-unit conversion
Velocity km/h 1 km/h โ‰ˆ 0.277778 m/s
Velocity mph 1 mph = 0.44704 m/s
Distance feet 1 ft = 0.3048 m
Distance yards 1 yd = 0.9144 m
Angle degrees degrees ร— ฯ€ / 180 = radians

Example Problems

Example 1: Find the range

You launch a projectile at an initial velocity of 20 m/s and an angle of 30ยฐ.

R = (20ยฒ ร— sin(2 ร— 30ยฐ)) / 9.81
R = (400 ร— sin(60ยฐ)) / 9.81 = 35.31 m

The range is about 35.31 m.

Example 2: Find the angle of projection

You know the projectile has an initial velocity of 25 m/s and a range of 40 m.

theta = arcsin((40 ร— 9.81) / 25ยฒ) / 2
theta = arcsin(0.62784) / 2 โ‰ˆ 19.4455ยฐ

The lower angle of projection is about 19.4455ยฐ. The complementary angle, about 70.5545ยฐ, gives the same range under ideal conditions.

FAQ

Why does 45 degrees give the maximum range?

The range formula contains sin(2ฮธ). The largest possible value of sine is 1, and that happens when 2ฮธ = 90ยฐ. Therefore, ฮธ = 45ยฐ gives the maximum range when launch height and landing height are the same and air resistance is ignored.

Why can two different angles give the same range?

Angles that add to 90 degrees have the same range in ideal projectile motion. For example, 30ยฐ and 60ยฐ give the same value of sin(2ฮธ), so they produce the same horizontal range if the initial velocity is the same.

When is there no valid angle?

There is no valid angle if the requested range is too large for the given initial velocity. In the formula, (R ร— g) / vยฒ must be between 0 and 1 for this nonnegative-range model. Angles are restricted to 0ยฐโ€“90ยฐ. Zero speed cannot determine a unique angle; zero range at an endpoint angle cannot determine a unique speed.