Find the initial velocity of $P$.

Find the value of $p$.

(i)     Find the value of $q$.

(ii)     Hence, find the speed of P when $t = q$.

(i)     Find the total distance travelled by P between $t = 1$ and $t = p$.

(ii)     Hence or otherwise, find the displacement of P from A when $t = p$.

valid attempt to substitute $t = 0$ into the correct function     (M1)

eg$\,\,\,\,\,$$- 2(0) + 2$

2     A1     N2

[2 marks]

recognizing $v = 0$ when P is at rest     (M1)

5.21834

$p = 5.22{\text{ }}({\text{seconds}})$     A1     N2

[2 marks]

(i)     recognizing that $a = v'$     (M1)

eg$\,\,\,\,\,$$v' = 0$, minimum on graph

1.95343

$q = 1.95$     A1     N2

(ii)     valid approach to find their minimum     (M1)

eg$\,\,\,\,\,$$v(q),{\text{ }} - 1.75879$, reference to min on graph

1.75879

speed $= 1.76{\text{ }}(c\,{\text{m}}\,{{\text{s}}^{ - 1}})$     A1     N2

[4 marks]

(i)     substitution of correct $v(t)$ into distance formula,     (A1)

eg$\,\,\,\,\,$$\int_1^p {\left| {3\sqrt t  + \frac{4}{{{t^2}}} - 7} \right|{\text{d}}t,{\text{ }}\left| {\int {3\sqrt t  + \frac{4}{{{t^2}}} - 7{\text{d}}t} } \right|}$

4.45368

distance $= 4.45{\text{ }}({\text{cm}})$     A1     N2

(ii)     displacement from $t = 1$ to $t = p$ (seen anywhere)     (A1)

eg$\,\,\,\,\,$$- 4.45368,{\text{ }}\int_1^p {\left( {3\sqrt t  + \frac{4}{{{t^2}}} - 7} \right){\text{d}}t}$

displacement from $t = 0$ to $t = 1$     (A1)

eg$\,\,\,\,\,$$\int_0^1 {( - 2t + 2){\text{d}}t,{\text{ }}0.5 \times 1 \times 2,{\text{ 1}}}$

valid approach to find displacement for $0 \leqslant t \leqslant p$     M1

eg$\,\,\,\,\,$$\int_0^1 {( - 2t + 2){\text{d}}t + \int_1^p {\left( {3\sqrt t  + \frac{4}{{{t^2}}} - 7} \right){\text{d}}t,{\text{ }}\int_0^1 {( - 2t + 2){\text{d}}t - 4.45} } }$

$- 3.45368$

displacement $=  - 3.45{\text{ }}({\text{cm}})$     A1     N2

[6 marks]