# Projectile Range

## Motion in two dimensions - calculate the range of a projectile

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/s^{2}, 32.174 ft/s^{2})*

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}^{2} sin(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/s^{2})*

* = 1.27 s *

The maximum elevation of the ball can be calculated as

*h = 1/2 (9.81 m/s^{2}) (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/s^{2}) *

* = 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.

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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