18M.2.AHL.TZ2.H_7

pestleMathematicsAIHLPaper 218M· ahl-5-13-kinematic-problemssource ↗

A point P moves in a straight line with velocity $v$ ms⁻¹ given by $v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}$ at time t seconds, where t ≥ 0.

Determine the first time t₁ at which P has zero velocity.

[2]

Find an expression for the acceleration of P at time t.

[2]

b.i.

Find the value of the acceleration of P at time t₁.

[1]

b.ii.

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to solve $v\left( t \right) = 0$ for t or equivalent (M1)

t₁ = 0.441(s) A1

[2 marks]

$a\left( t \right) = \frac{{{\text{d}}v}}{{{\text{d}}t}} = - {{\text{e}}^{ - t}} - 16t{{\text{e}}^{ - 2t}} + 16{t^2}{{\text{e}}^{ - 2t}}$ M1A1

Note: Award M1 for attempting to differentiate using the product rule.

[2 marks]

b.i.

$a\left( {{t_1}} \right) = - 2.28$ (ms⁻²) A1

[1 mark]

b.ii.

Examiners’ report

[N/A]

b.i.

[N/A]

b.ii.