The SUVAT equations apply only to motion with constant (uniform) acceleration

Motion & Kinematics - OCR A-Level Physics

When can I use SUVAT?

SUVAT applies only to motion with constant (uniform) acceleration in a straight line. Free fall (ignoring air resistance) qualifies. A skydiver before terminal velocity does not, because air resistance changes with speed.

The five variables

$s$ = displacement ($\text{m}$). $u$ = initial velocity ($\text{m s}^{-1}$). $v$ = final velocity ($\text{m s}^{-1}$). $a$ = acceleration ($\text{m s}^{-2}$). $t$ = time ($\text{s}$). You need three known values to find a fourth.

$$v = u + at$$

s = ut + \frac{1}{2}at^{2}

v^{2} = u^{2} + 2as

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

Case 1: Dropped from rest

$u = 0$. Useful equation: $s = \tfrac{1}{2}at^2$. A stone dropped from $20 \text{ m}$ has $t = \sqrt{\frac{2 \times 20}{9.81}} = 2.02 \text{ s}$.

Case 2: Brought to rest

$v = 0$. For a car braking from $20 \text{ m s}^{-1}$ to rest over $50 \text{ m}$: use $v^2 = u^2 + 2as$. $0 = 400 + 2a(50)$ gives $a = -4.0 \text{ m s}^{-2}$ (negative because it decelerates).

Case 3: Thrown upward

Choose upward as positive. Then $u > 0$ and $a = -9.81 \text{ m s}^{-2}$ throughout the flight (gravity acts downward). At the highest point, $v = 0$. Maximum height: $0 = u^2 + 2(-9.81)h \Rightarrow h = \frac{u^2}{2g}$.

Case 4: Multi-stage motion

Each stage has its own constant acceleration. Apply SUVAT to each stage separately. The $v$ at the end of one stage becomes the $u$ at the start of the next.

"Starts from rest" $\Rightarrow u = 0$. "Comes to rest" $\Rightarrow v = 0$. "Constant velocity" $\Rightarrow a = 0$ (and SUVAT collapses to $s = vt$).
Choosing the right equation: pick the one that does not contain the variable you don't know and don't want. Need $s$, know $u, a, t$? Use $s = ut + \tfrac{1}{2}at^2$ (no $v$).
If the problem has two unknowns, you usually need two SUVAT equations or one SUVAT plus another physics equation (force, energy).

Common Mistake HIGH

Wrong: Applying SUVAT to motion where acceleration is not constant (a falling skydiver, a car with changing throttle), or mixing sign conventions partway through a question.
Right: Confirm uniform acceleration before reaching for SUVAT. State your sign convention at the start ("take upward as positive") and stick with it. If the object moves up then back down, $a = -9.81 \text{ m s}^{-2}$ for the whole flight.

Examiner Tips and Tricks

Always list your known SUVAT variables before choosing an equation: write $s = \ldots, u = \ldots, v = \ldots, a = \ldots, t = \ldots$, mark the unknown with a "?".
"Starts from rest" $\Rightarrow u = 0$. "Comes to rest" $\Rightarrow v = 0$. "Released" or "dropped" $\Rightarrow u = 0$.
OCR mark schemes accept any valid SUVAT route. Pick the one with fewest steps.

Worked Example [5 marks]

A car accelerates from rest at 2.0 $m s^{-2}$ for 10 s, then brakes at -4.0 $m s^{-2}$ until it stops. Find the total distance travelled.

Show Solution

Stage 1

$u = 0, a = 2.0 \text{ m s}^{-2}, t = 10 \text{ s}. Use v = u + at: v = 0 + 2.0 \times 10 = 20 \text{ m s}^{-1}$

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Stage 1 distance

$s_1 = ut + \frac{1}{2}at^{2} = 0 + \frac{1}{2}(2.0)(10^{2}) = 100 \text{ m}$

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Stage 2

$u = 20, v = 0, a = -4.0. Use v^{2} = u^{2} + 2as: 0 = 400 + 2(-4.0)s$

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$s_2 = \frac{400}{8.0} = 50 \text{ m}$

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$Total distance = s_1 + s_2 = 100 + 50 = 150 \text{ m}$

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Answer

150 m