Projectile Motion

Decompose the Launch

across and up are separate

A projectile is two simple motions at once: a steady horizontal coast and a vertical throw under gravity. Splitting them is the whole trick.

Example

A projectile is two simple motions at once: a steady horizontal coast and a vertical throw under gravity. Splitting them is the whole trick.

highlighted = computed this step

A ball thrown up and forward

A ball is launched with a horizontal part of 5 metres per second and a vertical part of 10 metres per second. The slanted launch arrow is the real velocity; we work in these two parts precisely so we never need its messy diagonal length. The trick is to treat across and up as two separate, simpler problems.

vx=5 m/s,vy=10 m/sv_x = 5\ \text{m}/\text{s}, \quad v_y = 10\ \text{m}/\text{s}
The launch velocity split into across and upThe launch arrow with its constant horizontal part and its upward vertical part drawn from the launch point.ballvxvylaunch

Across: nothing pushes it sideways

Gravity only pulls down, so nothing changes the across speed. The horizontal part stays 5 metres per second the whole flight: across position is just that speed times time, equal steps every second.

x=vx t=5 m/s ⋅ tx = v_x\,t = 5\ \text{m}/\text{s}\,\cdot\,t
Across alone: equal steps every secondThe sideways position by itself: equally spaced marks with equal arrows, because the across speed never changes.t=0t=1t=2

Up: gravity slows it, stops it, drops it

The up part is free fall with a head start. Gravity takes 10 metres per second off the upward speed each second, so the climb slows to a stop at the top, then falls back. The vertical position is the launch-up minus the usual one-half g t squared.

y=vy t−12 g t2y = v_y\,t - \tfrac{1}{2}\,g\,t^{2}
Up alone: the climb slows toward the topThe upward position by itself on a vertical line: the gaps shrink as gravity slows the climb to a stop at the top.starttop
mechanics Giving the launch as a clean horizontal part and vertical part lets each direction be solved on its own.