Projectile Range

The time for a projectile - a bullet, a ball or a stone or something similar - thrown out with an angle Θ to the horizontal plane - to reach the maximum height can be calculated as

t_h = v_i sin(Θ) / a_g (1)

where

t_h = time to reach maximum height (s)

v_i = initial velocity of the projectile (m/s, ft/s)

Θ = the initial angle of the velocity vector to the horizontal plane (degrees)

a_g = acceleration of gravity (9.81 m/s², 32.174 ft/s²)

The total flight time can be expressed as

t_s = 2 t_h

= 2 v_i sin(Θ) / a_g (2)

where

t_s = time for the total flight (s)

The maximum elevation - h - of the flight can be calculated  as

h = 1/2 a_g t_h ² (3)

where

h = flight maximum elevation (m, ft)

The horizontal distance of the flight can be expressed as

s = v_i ²sin(2 Θ ) / a_g (4)

where

s = flight distance (m, ft)

Example - Throwing a Ball

A ball is thrown with initial velocity 25 m/s in angle 30 degrees to the horizontal plane. The time for the ball to reach maximum level is

t_h = ( 25 m/s ) sin(30 degrees) / ( 9.81 m/s²)

= 1.27 s

The maximum elevation of the ball can be calculated as

h = 1/2 ( 9.81 m/s²) ( 1.27 s < )²

= 7.91 m

The horizontal distance traveled by the ball can be calculated as

s = ( 25 m/s )²sin(2 30 degrees ) / ( 9.81 m/s²)

= 55.2 m

Projectile Flight Calculator

The calculator below can be used to estimate the time of flight, time to reach maximum elevation, maximum elevation and flight distance for a projectile like a bullet or a ball or something similar.

Note that friction force due to air resistance is neglected.

v_i - initial velocity (m/s)

Θ - initial angle to the horizontal plane (degrees)

• Load Calculator!

Note that friction force due to air resistance is neglected in the charts above. For higher velocities range and maximum height will be dramatically reduced.