# Range of a Projectile

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

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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 forces due to air resistance is neglected.

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