21M.2.SL.TZ1.5

pestleMathematicsAASLPaper 221M· sl-5-9-kinematics-problemssource ↗

A particle moves in a straight line. The velocity, $v {ms}^{- 1}$ , of the particle at time $t$ seconds is given by $v (t) = t \sin t - 3$ , for $0 \leq t \leq 10$ .

The following diagram shows the graph of $v$ .

Find the smallest value of $t$ for which the particle is at rest.

[2]

Find the total distance travelled by the particle.

[2]

Find the acceleration of the particle when $t = 7$ .

[2]

Markscheme / solution

recognising $v = 0$ (M1)

$t = 6.74416 \dots$

$= 6.74$ (sec) A1

Note: Do not award A1 if additional values are given.

[2 marks]

$\int_{0}^{10} |v (t)| d t$ OR $- \int_{0}^{6.74416 \dots} v (t) d t + \int_{6.74416 \dots}^{9.08837 \dots} v (t) d t - \int_{9.08837 \dots}^{10} v (t) d t$ (A1)

$= 37.0968 \dots$

$= 37.1 (m)$ A1

[2 marks]

recognizing acceleration at $t = 7$ is given by $v' (7)$ (M1)

acceleration $= 5.93430 \dots$

$= 5.93 ({ms}^{- 2})$ A1

[2 marks]

Examiners’ report

[N/A]