Find the particle’s acceleration in terms of $s$.

Using an appropriate sketch graph, find the particle’s displacement when its acceleration is $0.25{\text{ m}}{{\text{s}}^{ - 2}}$.

$\frac{{{\text{d}}v}}{{{\text{d}}s}} = \frac{{\cos s}}{{{{\sin }^2}s + 1}}$    M1A1

$a = v\frac{{{\text{d}}v}}{{{\text{d}}s}}$    (M1)

$a = \frac{{\arctan (\sin s)\cos s}}{{{{\sin }^2}s + 1}}$    A1

[4 marks]

EITHER

     (M1)

OR

     (M1)

THEN

$s = 0.296,{\text{ }}0.918{\text{ (m)}}$     A1

[2 marks]

In part (a), a large number of candidates thought that $\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{{{\text{d}}v}}{{{\text{d}}s}}$ rather than $\frac{{{\text{d}}v}}{{{\text{d}}t}} = \frac{{{\text{d}}s}}{{{\text{d}}t}} \times \frac{{{\text{d}}v}}{{{\text{d}}s}} = v\frac{{{\text{d}}v}}{{{\text{d}}s}}$.

In part (b), quite a few of these candidates then went on to find a value of $s$ that was outside the domain $0 \leqslant s \leqslant 1$.