3.4.1.3MechanicsCore

Motion and the SUVAT Equations

Kinematics is the description of motion using displacement, velocity and acceleration. When acceleration is $constant$ , four equations — the SUVAT equations — connect these quantities and let you solve almost any straight-line motion problem.

40 min Video by Science Shorts 3.4.1 Force, energy and momentum

SUVAT — AS/A-Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Define displacement, velocity and acceleration
Interpret displacement–time and velocity–time graphs (gradient and area)
Recall and select the correct SUVAT equation for a problem
Solve problems involving uniform acceleration, including free fall
Apply a consistent sign convention to motion in a straight line

Describing motion

$Velocity$ is the rate of change of displacement, and $acceleration$ is the rate of change of velocity. Both are vectors, so direction matters and a negative value means motion (or acceleration) in the opposite direction.

A useful definition to keep straight: average velocity is total displacement over total time, which is not always the same as average speed.

v = \frac{Δ s}{Δ t}, a = \frac{Δ v}{Δ t}

Velocity is the rate of change of displacement; acceleration is the rate of change of velocity.

Motion graphs

Graphs encode motion compactly. On a $displacement-time$ graph the gradient is the velocity. On a $velocity-time$ graph the gradient is the acceleration and the $area under the line$ is the displacement.

These two facts — gradient and area — turn many exam questions into simple geometry.

Tip — On a velocity–time graph: a straight sloped line is constant acceleration, a horizontal line is constant velocity, and area below the time axis counts as negative (backward) displacement.

The SUVAT equations

When acceleration is constant, the five quantities $s$ (displacement), $u$ (initial velocity), $v$ (final velocity), $a$ (acceleration) and $t$ (time) are linked by the four equations below. Each equation is missing exactly one of the five variables.

The strategy is always the same: write down the three quantities you know and the one you want, then pick the equation that contains those four and not the fifth.

v = u + a t

s

s = u t + \frac{1}{2} a t^{2}

v

v^{2} = u^{2} + 2 a s

t

s = \frac{1}{2} (u + v) t

a

1Known:

u = 0

a = 3.0

m/s

^{2}

t = 5.0

s. Want

s

2Use

s = u t + \frac{1}{2} a t^{2}

(no

v

needed).

s = 0 + \frac{1}{2} (3.0) (5.0)^{2} = 37.5

Answer

s = 37.5

Equation recap

v = u + a t

Velocity after time t (no s).

s = u t + \frac{1}{2} a t^{2}

Displacement (no v).

v^{2} = u^{2} + 2 a s

Links v and s (no t).

s = \frac{1}{2} (u + v) t

Displacement from average velocity (no a).

Common mistakes to avoid

Using a SUVAT equation when the acceleration is not constant.

SUVAT only applies to uniform acceleration. For changing acceleration, use a graph (gradient/area) instead.

Taking g as positive while calling "up" positive, then getting the wrong sign.

Choose one direction as positive and stick to it. If up is positive, then g acts downward and a = −9.81 m/s².

Forgetting that an object dropped or starting from rest has u = 0.

"From rest" means the initial velocity u = 0 — write it down explicitly.

Key takeaways

Velocity is rate of change of displacement; acceleration is rate of change of velocity.
On a v–t graph, gradient = acceleration and area = displacement.
The four SUVAT equations apply only when acceleration is constant.
Pick the equation containing your three knowns and the unknown you want.

Test yourself

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