Projectile motion

Physics · Motion and force. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A projectile is a body thrown into the air that then moves only under gravity, with air resistance ignored. Its path is a parabola. The idea to hold is that the horizontal and vertical motions are independent.

Horizontally there is no force, so the horizontal velocity stays constant for the whole flight. Vertically the acceleration is g downwards, so the vertical velocity falls to zero at the highest point and then grows downwards. At the top the projectile still moves horizontally and its acceleration is still g.

Time links the two motions: find the time from the vertical motion, then use it in the horizontal motion.

Rules to remember

Components of the launch velocity: v_ix = v_i cos θ and v_iy = v_i sin θ.
Time of flight: t = 2v_i sin θ ÷ g.
Maximum height: h = v_i² sin²θ ÷ (2g).
Horizontal range: R = v_i² sin 2θ ÷ g.
Range is greatest at θ = 45°, where R_max = v_i² ÷ g.
Complementary angles such as 30° and 60° give the same range at the same speed.
Thrown horizontally from height h: time to fall t = √(2h ÷ g); distance = v_x × t.

Common mistake

Students say the velocity is zero at the highest point. Only the vertical component is zero there; the horizontal component is unchanged and the acceleration is still 9.8 m s⁻² downwards.

Worked example

A ball is kicked from level ground at 28 m s⁻¹ at 30° above the horizontal. Find the time of flight, the maximum height and the range. (g = 9.8 m s⁻²)

t = 2v_i sin θ ÷ g = 2 × 28 × 0.5 ÷ 9.8 = 28 ÷ 9.8 = 2.86 s.
h = v_i² sin²θ ÷ (2g) = (28 × 0.5)² ÷ 19.6 = 196 ÷ 19.6 = 10 m.
R = v_i² sin 2θ ÷ g = 784 × sin 60° ÷ 9.8 = 80 × 0.866 = 69.3 m.

Answer: Time of flight 2.86 s, maximum height 10 m, range 69.3 m

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