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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

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.

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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Related Topics

  • Dynamics

    Motion of bodies and the action of forces in producing or changing their motion - velocity and acceleration, forces and torque.
  • Mechanics

    The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

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