Projectile Motion — Quick Summary

Mr Toogood's Physics · Mechanics

AQA 3.4.1.4
u_H=u cosθ, u_V=u sinθ
Resolving launch velocity
s_H = u_H t
Range
s_V=u_Vt+½at²
Vertical displacement
t = √(2s/g)
Horizontal launch only

Independence of motion

A projectile moves freely under gravity alone, with no continuous thrust. Its horizontal and vertical motions are completely independent — neither affects the other.

VerticalHorizontal
Acceleration−g0
Timet (same for both)

Solving method

  1. If launched at an angle, resolve u into u_H=u cosθ and u_V=u sinθ first.
  2. List the knowns separately for horizontal and vertical motion.
  3. Solve the vertical motion first — this usually gives the time of flight, t.
  4. Use that t in the horizontal equation, s_H=u_Ht, for range or other horizontal quantities.

Launched horizontally

Diagram of a horizontally launched projectile, vertical velocity growing while horizontal velocity stays constant

Vertical velocity grows from zero; horizontal velocity stays constant throughout.

With u_V=0, time of flight depends only on the height fallen:

t = √(2s/g)
Independent of speed: two objects launched horizontally at different speeds from the same height land at exactly the same time — only their range differs.

The monkey and the hunter

The monkey and hunter thought experiment showing the bullet's curved path meeting the monkey's vertical fall

Gravity accelerates the bullet and the falling monkey identically — the bullet always hits.

A bullet aimed straight at a monkey always hits it, even if the monkey drops the instant the gun fires — because gravity accelerates both equally and identically. This works precisely because horizontal and vertical motion are independent, regardless of the bullet's speed or launch angle.

Air resistance

Comparison of a projectile's path with and without air resistance, the path with resistance falling short and steeper

Air resistance reduces range and height, and steepens the final descent.

Drag increases with speed and opposes motion, reducing both velocity components:

  • Lower maximum height, reached sooner.
  • Shorter range.
  • Steeper descent than ascent — the path is no longer symmetrical.

Terminal velocity is reached when drag exactly balances weight (no more acceleration) — the same principle limits the top speed of powered vehicles, where drag balances the driving force.

Exam essentials

Key equations

  • u_H=u cosθ, u_V=u sinθ
  • s_H=u_Ht
  • s_V=u_Vt+½at²
  • t=√(2s/g) (horizontal launch)

Useful symmetry trick

  • Time to rise to max height = time to fall back to launch height.
  • Find time to the top (v=0) using v=u+at, then double it.

Common slips

  • Horizontal launch time depends only on height, never on speed.
  • Solve vertical motion first — it's usually where t comes from.
  • Air resistance breaks the parabola's symmetry — steeper on the way down, don't assume it mirrors the ascent.