SPM.1.AHL.TZ0.7

pestleMathematicsAIHLPaper 1SPM· ahl-5-9-differentiating-standard-functions-and-derivative-rulessource ↗

A particle, A, moves so that its velocity ( $\nu$ ms⁻¹) at time $t$ is given by $\nu$ = 2 sin $t$ , $t$ ≥ 0.

The kinetic energy ( $E$ ) of the particle A is measured in joules (J) and is given by $E$ = 5 $\nu$ ².

Write down an expression for $E$ as a function of time.

[1]

Hence find $\frac{{{\text{d}}E}}{{{\text{d}}t}}$ .

[2]

Hence or otherwise find the first time at which the kinetic energy is changing at a rate of 5 J s⁻¹.

[2]

Markscheme / solution

$E = 5{\left( {2\,{\text{sin}}\,t} \right)^2}\,\left( { = 20\,{\text{si}}{{\text{n}}^2}\,t} \right)$ A1

[1 mark]

$\frac{{{\text{d}}E}}{{{\text{d}}t}} = 40\,{\text{sin}}\,t\,{\text{cos}}\,t$ (M1)A1

[2 marks]

$t$ = 0.126 (M1)A1

[2 marks]

Examiners’ report

[N/A]