Find an expression for the velocity, $v$, of the particle at time $t$.

Find an expression for the acceleration, $a$, of the particle at time $t$.

Find the acceleration of the particle at time $t = 0$.

$v - \frac{{{\text{d}}s}}{{{\text{d}}t}} = \frac{{{{\text{e}}^{ - t}}}}{{2 - {{\text{e}}^{ - t}}}}{\text{ }}\left( { = \frac{1}{{2{{\text{e}}^t} - 1}}{\text{ or }} - 1 + \frac{2}{{2 - {{\text{e}}^{ - t}}}}} \right)$     M1A1

[2 marks]

$a = \frac{{{{\text{d}}^2}s}}{{{\text{d}}{t^2}}} = \frac{{ - {{\text{e}}^{ - t}}(2 - {{\text{e}}^{ - t}}{\text{)}} - {{\text{e}}^{ - t}} \times {{\text{e}}^{ - t}}}}{{{{(2 - {{\text{e}}^{ - t}})}^2}}}{\text{ }}\left( { = \frac{{ - 2{{\text{e}}^{ - t}}}}{{{{(2 - {{\text{e}}^{ - t}})}^2}}}} \right)$     M1A1

Note:     If simplified in part (a) award (M1)A1 for $a = \frac{{{{\text{d}}^2}s}}{{{\text{d}}{t^2}}} = \frac{{ - 2{{\text{e}}^t}}}{{{{(2{{\text{e}}^t} - 1)}^2}}}$.

Note:     Award M1A1 for $a =  - {{\text{e}}^{ - t}}{(2 - {{\text{e}}^{ - t}})^{ - 2}}({{\text{e}}^{ - t}}) - {{\text{e}}^{ - t}}{(2 - {{\text{e}}^{ - t}})^{ - 1}}$.

[2 marks]

$a =  - 2{\text{ }}({\text{m}}{{\text{s}}^{ - 2}})$     A1

[1 mark]

Mostly well done. There were a few sign errors but most candidates were correctly applying the quotient or chain rules.

Mostly well done. There were a few sign errors but most candidates were correctly applying the quotient or chain rules.

Mostly well done. There were a few sign errors but most candidates were correctly applying the quotient or chain rules.