Equations of Motion

The question

A jet must reach 80 m/s to lift off and the runway is only 1500 m long. You know the engine's acceleration and the takeoff speed — but nobody handed you the time. Can you still prove the plane gets airborne in time?

When acceleration is constant, four equations link u, v, a, s and t — and each one skips a different quantity. Pick the equation missing the variable you don't have, and any constant-acceleration problem cracks open.

The key idea

When acceleration is constant, five quantities describe the motion: initial velocity u, final velocity v, acceleration a, displacement s, and time t. Four 'SUVAT' equations connect them, and each equation leaves out exactly one quantity. To solve any problem, list what you know, spot the one variable you neither have nor want, and choose the equation that omits it.

The trick is matching equation to unknowns. No time given? Use $v^2 = u^2$ + 2as. Don't know the final velocity yet? Use s = ut + ½at². The equation $s = \tfrac{1}{2}(u+v)t$ is just average velocity ½(u+v) multiplied by time — handy when a isn't given. Watch signs: pick a positive direction and keep u, v, a and s consistent with it; a deceleration is simply a negative a. For vertical motion under gravity, $a = -9.8\,\text{m/s}^2$ (the topic of the next lesson, Free Fall). **Limiting case:** set $a = 0$ and all three equations collapse to $s = vt$; set $u = 0$ and they simplify to $v = at$, $s = \tfrac{1}{2}at^2$ — sanity-check every answer against these. **Connect it:** all three live inside the v–t graph: $v = u + at$ is the straight line itself, $s$ is the area under it, and $v^2 = u^2 + 2as$ is what remains when you eliminate $t$ between them. One picture, three equations.

The formula

• ·u = initial velocity (m/s)
• ·v = final velocity (m/s)
• ·a = acceleration (m/s²)
• ·s = displacement (m)
• ·t = time (s). v = u + at omits s; s = ut + ½at² omits v; v² = u² + 2as omits t; s = ½(u+v)t omits a. All assume a is constant.

What to remember

• ·The four SUVAT equations link u, v, a, s, t; each one leaves a different variable out.
• ·They apply only to CONSTANT acceleration.
• ·Pick the equation containing your three knowns plus the unknown.