projectile

When there is no air resistance, the trajectory, the path traced out by the projectile, is a parabola whose vertex corresponds to the point at which the projectile attains its maximum height. Take the origin at the point of projection, the x-axis horizontal and the y-axis vertical with the positive direction upwards. Then the equation of motion is , subject to r = 0 and at t = 0, where v is the speed of projection and θ is the angle of projection. This gives

for t ≥ 0. It follows that y = 0 when t =  (2v/g)sinθ, the time of flight, and x = (v2/g)sinθ, the range of the projectile.

The maximum height of v2sin2θ/(2g) is attained halfway through the flight. The maximum range, for any given value of v, is obtained when θ=π/4. Similar investigations can be carried out for a projectile projected from a point a given height above a horizontal plane or projected up or down an inclined plane. For a body projected vertically upwards with initial speed v, set θ = π/2.

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