Find the velocity of the particle when \(t = 1\).

Find the value of \(t\) for which the particle is at rest.

Find the total distance the particle travels during the first three seconds.

Show that the acceleration of the particle is given by \(a = 6t{({t^2} - 4)^2}\).

Find all possible values of \(t\) for which the velocity and acceleration are both positive or both negative.

substituting \(t = 1\) into \(v\)     (M1)

eg     \(v(1),{\text{ }}{\left( {{1^2} - 4} \right)^3}\)

velocity \( =  - 27{\text{ }}\left( {{\text{m}}{{\text{s}}^{ - 1}}} \right)\)     A1     N2

[2 marks]

valid reasoning     (R1)

eg     \(v = 0,{\text{ }}{\left( {{t^2} - 4} \right)^3} = 0\)

correct working     (A1)

eg     \({t^2} - 4 = 0,{\text{ }}t =  \pm 2\), sketch

\(t = 2\)     A1     N2

[3 marks]

correct integral expression for distance     (A1)

eg     \(\int_0^3 {\left| v \right|,{\text{ }}\int {\left| {{{\left( {{t^2} - 4} \right)}^3}} \right|,{\text{ }} - \int_0^2 {v{\text{d}}t + \int_2^3 {v{\text{d}}t} } } } \),

\(\int_0^2 {{{\left( {4 - {t^2}} \right)}^3}{\text{d}}t + \int_2^3 {{{\left( {{t^2} - 4} \right)}^3}{\text{d}}t} }\) (do not accept \(\int_0^3 {v{\text{d}}t} \))

\(86.2571\)

\({\text{distance}} = 86.3{\text{ (m)}}\)     A2     N3

[3 marks]

evidence of differentiating velocity     (M1)

eg     \(v'(t)\)

\(a = 3{\left( {{t^2} - 4} \right)^2}(2t)\)     A2

\(a = 6t{\left( {{t^2} - 4} \right)^2}\)     AG     N0

[3 marks]

METHOD 1

valid approach     M1

eg     graphs of \(v\) and \(a\)

correct working     (A1)

eg     areas of same sign indicated on graph

\(2 < t \leqslant 3\)   (accept \(t > 2\))     A2     N2

METHOD 2

recognizing that \(a \geqslant 0\) (accept \(a\) is always positive) (seen anywhere)     R1

recognizing that \(v\) is positive when \(t > 2\) (seen anywhere)     (R1)

\(2 < t \leqslant 3\)   (accept \(t > 2\))     A2     N2

[4 marks]