Constant-acceleration equations | CAIE 9709 Mechanics Notes

Mechanics

Mathematics9709

Confirm constant acceleration before using suvat

v=u+at

Symbols

$u$ = velocity at the start of this stage (m s⁻¹)
$v$ = velocity at the end of this stage (m s⁻¹)
$a$ = constant acceleration (m s⁻²)
$t$ = duration of this stage (s)

Choose a positive direction before assigning signs.
$s$ means displacement during the stage, not total distance.
$t$ means the duration of the stage, unless a global time is explicitly used.
Do not use these equations when acceleration is a function of time.

Common mistake

“Decelerates at 3” describes a speed decrease, not automatically

a=-3

. The sign depends on the chosen positive direction and the particle's direction of motion.

Choose the equation that omits the unwanted variable

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List the known quantities first. Choose the equation that leaves out the variable you neither know nor need.

v=u+at

s=ut+\tfrac12at^2

s=vt-\tfrac12at^2

s=\tfrac12(u+v)t

v^2=u^2+2as

Write a five-letter data line first. The equation choice should then be mechanical, not guessed from memory.

Solve the algebra, then test the physical meaning

Worked example

Two positive roots can both describe the motion

A particle has $u=5$ , $a=-2$ and travels $s=6$ . Using $s=ut+\tfrac12at^2$ gives $t^2-5t+6=0$ , so $t=2$ or $t=3$ .

Both times can be valid: the particle passes the same position before and after turning. Check the velocity and the question's stated interval before rejecting a root.

Reject a negative time when time is measured after the start.
Do not reject a negative velocity merely because speed is positive.
Check units and whether the answer matches the chosen direction.
Keep unrounded values until the final answer.

Examiner note

Once the motion equation is written, a calculator equation solver is often the fastest main route to its numerical roots. If it factors into convenient numbers, show that factorisation. If the roots are awkward, enter the equation directly. In both cases, record the equation and every root, then test each one against the stated time interval and direction. The calculator does not replace the motion model.

Your turnIndependent check [5 marks]

A train moves in a straight line with constant acceleration $0.6\text{ m s}^{-2}$ . Its initial velocity is $4\text{ m s}^{-1}$ and it travels 90 m. Find its final velocity and the time taken.

Show worked answer

Time is unwanted at first, so use the equation that omits it.

v^2=4^2+2(0.6)(90)=124

Initial velocity and acceleration are both positive, so velocity stays positive during this stage. Take the positive square root.

v=\sqrt{124}=11.1\text{ m s}^{-1}

11.1355=4+0.6t

t=11.9\text{ s}

Answer

v=11.1\text{ m s}^{-1}

t=11.9\text{ s}