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Projectile Range

Calculate the range of a projectile - a motion in two dimensions.

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Projectile - flight and 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

th = vi sin(Θ) / ag                                 (1)


th = time to reach maximum height (s)

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

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

ag = acceleration of gravity (9.81 m/s232.174 ft/s2)

The total flight time can be expressed as

ts = 2 th

    = 2 vi sin(Θ) / ag                      (2)


ts = time for the total flight (s)

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

h = 1/2 ag th2                              (3)


h = flight maximum elevation (m, ft)

The horizontal distance of the flight can be expressed as

s = vi2 sin(2 Θ) / ag                           (4)


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

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

    = 1.27

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)2 sin(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.

vi - initial velocity (m/s)

Θ - initial angle to the horizontal plane (degrees)

Projectile - velocity, elevation and maximum distance

Projectile - velocity, elevation and maximum height

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.

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