A projectile is two simple motions happening together and never interfering: a steady horizontal velocity and a vertical fall under gravity. Split the launch into components, treat each with the tools you already have, and the curved path becomes two straight-line problems.
Horizontal and vertical motion are independent. Horizontally there is no force (ignoring air resistance), so the velocity is constant: x = (u cosθ) t. Vertically there is gravity, so the motion obeys the suvat equations with a = g: y = (u sinθ) t − ½ g t². Time is the link between the two.
Set a launch speed and angle and fire. Turn on the vertical twin, a ball thrown straight up at the same vertical speed but with no sideways motion. The dashed connector shows the projectile and the twin stay at the same height the whole way, and they land together, because gravity ignores horizontal motion. Predict when the twin lands before you check.
For a launch speed u at angle θ to the horizontal, split the velocity once at the start, then handle each direction with its own rules.
| Direction | Initial component | Behaviour |
|---|---|---|
| horizontal | u cosθ | constant velocity (a = 0) |
| vertical | u sinθ | uniform acceleration, a = g downward |
Useful results follow from these: time to the highest point when u sinθ = g t, full time of flight on level ground t = 2u sinθ / g, and range R = (u cosθ) t. The speed at any instant recombines the two components with Pythagoras.
Four quick checks on independence and components. Each correct answer earns XP and lights this skill on your star map.
A ball is thrown horizontally from a table at the same instant an identical ball is dropped from the table edge. Ignoring air resistance, they reach the floor:
During the flight of a projectile, ignoring air resistance, the horizontal component of its velocity:
A projectile is launched at 20 m s⁻¹ at 30° above the horizontal. Its initial vertical component of velocity is:
At the highest point of a projectile's path, its vertical velocity and vertical acceleration are:
Never mix the components. Gravity acts only vertically, so it never changes the horizontal velocity; do not apply g to the horizontal motion. For a ball launched horizontally, the time to land depends on the drop height alone, not on how fast it was thrown sideways: a faster throw simply lands further away in the same time. And remember the acceleration at the top of the arc is still g, not zero, even though the vertical velocity there is zero.
Unlocks once the four checks above are done. Worth more XP, written to AS Paper 1 and 2 standard.
A ball rolls off a table 1.25 m high and lands 2.0 m away horizontally. Taking g = 9.8 m s⁻², the time of flight is closest to:
For the same ball in the previous style of problem, with a time of flight of 0.50 s and a horizontal range of 2.0 m, the horizontal launch speed was:
A projectile is launched at 25 m s⁻¹ at 53° to the horizontal (sin 53° = 0.80). Taking g = 9.8 m s⁻², the time to reach its highest point is closest to:
Two balls are launched from ground level with the same speed, one at 30° and one at 60° to the horizontal. Ignoring air resistance, their horizontal ranges are:
This skill is now lit gold on your star map. You have finished the lessons of Topic 2; the Paper 1 set awaits.