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Date	May 2017	Marks available	2	Reference code	17M.2.SL.TZ1.S_7
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 1
Command term	Find	Question number	S_7	Adapted from	N/A

Question

A particle P moves along a straight line. Its velocity ${v_{\text{P}}}{\text{ m}}\,{{\text{s}}^{ - 1}}$ after $t$ seconds is given by ${v_{\text{P}}} = \sqrt t \sin \left( {\frac{\pi }{2}t} \right)$ , for $0 \leqslant t \leqslant 8$ . The following diagram shows the graph of ${v_{\text{P}}}$ .

M17/5/MATME/SP2/ENG/TZ1/07

Write down the first value of $t$ at which P changes direction.

[1]

a.i.

Find the total distance travelled by P, for $0 \leqslant t \leqslant 8$ .

[2]

a.ii.

A second particle Q also moves along a straight line. Its velocity, ${v_{\text{Q}}}{\text{ m}}\,{{\text{s}}^{ - 1}}$ after $t$ seconds is given by ${v_{\text{Q}}} = \sqrt t$ for $0 \leqslant t \leqslant 8$ . After $k$ seconds Q has travelled the same total distance as P.

Find $k$ .

[4]

b.

Markscheme

$t = 2$ A1 N1

[1 mark]

a.i.

substitution of limits or function into formula or correct sum (A1)

eg $\,\,\,\,\,$ $\int_0^8 {\left| v \right|{\text{d}}t,{\text{ }}\int {\left| {{v_Q}} \right|{\text{d}}t,{\text{ }}\int_0^2 {v{\text{d}}t - \int_2^4 {v{\text{d}}t + \int_4^6 {v{\text{d}}t - \int_6^8 {v{\text{d}}t} } } } } }$

9.64782

distance $= 9.65{\text{ (metres)}}$ A1 N2

[2 marks]

a.ii.

correct approach (A1)

eg $\,\,\,\,\,$ $s = \int {\sqrt t ,{\text{ }}\int_0^k {\sqrt t } } {\text{d}}t,{\text{ }}\int_0^k {\left| {{v_{\text{Q}}}} \right|{\text{d}}t}$

correct integration (A1)

eg $\,\,\,\,\,$ $\int {\sqrt t = \frac{2}{3}{t^{\frac{3}{2}}} + c,{\text{ }}\left[ {\frac{2}{3}{x^{\frac{3}{2}}}} \right]_0^k,{\text{ }}\frac{2}{3}{k^{\frac{3}{2}}}}$

equating their expression to the distance travelled by their P (M1)

eg $\,\,\,\,\,$ $\frac{2}{3}{k^{\frac{3}{2}}} = 9.65,{\text{ }}\int_0^k {\sqrt t {\text{d}}t = 9.65}$

5.93855

5.94 (seconds) A1 N3

[4 marks]

b.

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

b.

Syllabus sections

Topic 5—Calculus » AHL 5.13—Kinematic problems

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19M.2.SL.TZ2.S_8e.i:
Given that particles A and B start at the same point, find the displacement function ${s_{\text{B}}}$ for particle B.
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Find the value of $t$ when particle A first changes direction.
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Write down the first value of $t$ at which P changes direction.
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Find the magnitude of the particle’s acceleration at $6$ seconds.
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21N.1.AHL.TZ0.8b:
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22M.1.AHL.TZ2.17a:
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Find the total distance travelled in the first 5 seconds.
18M.2.SL.TZ2.S_9e:
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Find the initial velocity of P.
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Find the maximum speed of P.
18M.2.AHL.TZ2.H_7b.ii:
Find the value of the acceleration of P at time t₁.
17M.2.SL.TZ2.S_7:
Note: In this question, distance is in metres and time is in seconds.

A particle moves along a horizontal line starting at a fixed point A. The velocity $v$ of the particle, at time $t$ , is given by $v(t) = \frac{{2{t^2} - 4t}}{{{t^2} - 2t + 2}}$ , for $0 \leqslant t \leqslant 5$ . The following diagram shows the graph of $v$

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Find when the particle is at rest.
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Hence or otherwise, find all possible values of $t$ for which the velocity of P is decreasing.
17M.2.SL.TZ1.S_7b:
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Find the greatest speed of $P$ in the interval $0 \leq t \leq 6$ .
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Find the total distance travelled by the particle.
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Find an expression for the velocity of P at time $t$ .
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Write down the number of times that the acceleration of P is 0 m s⁻² .
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19M.2.AHL.TZ2.H_6:
A particle moves along a horizontal line such that at time $t$ seconds, $t$ ≥ 0, its acceleration $a$ is given by $a$ = 2 $t$ − 1. When $t$ = 6 , its displacement $s$ from a fixed origin O is 18.25 m. When $t$ = 15, its displacement from O is 922.75 m. Find an expression for $s$ in terms of $t$ .
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16N.2.SL.TZ0.S_9d:
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Find the magnitude of the velocity of the ice-skater when $t = 2$ .
21M.2.AHL.TZ1.6e:
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Find $OP$ .
21M.2.AHL.TZ1.6c:
Find the value of $a$ and the value of $b$ .
19M.1.SL.TZ1.S_7a:
Find the value of $s\left( 4 \right) - s\left( 2 \right)$ .
17N.1.AHL.TZ0.H_5:
A particle moves in a straight line such that at time $t$ seconds $(t \geqslant 0)$ , its velocity $v$ , in ${\text{m}}{{\text{s}}^{ - 1}}$ , is given by $v = 10t{{\text{e}}^{ - 2t}}$ . Find the exact distance travelled by the particle in the first half-second.
17M.2.AHL.TZ1.H_11a:
Find his velocity when $t = 15$ .
18M.2.AHL.TZ2.H_7b.i:
Find an expression for the acceleration of P at time t.
19M.2.SL.TZ2.S_8e.ii:
Find the other value of $t$ when particles A and B meet.
21M.2.AHL.TZ2.6a:
Find the times when $P$ is at instantaneous rest.
21M.2.AHL.TZ2.6e:
Find the total distance travelled by $P$ in the interval $0 \leq t \leq 4$ .
17M.1.AHL.TZ2.H_4b:
Find the displacement of the particle when $t = {t_1}$
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Find an expression for the velocity of the ball at time $t$ .
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Find the velocity vector at time $t$ .
EXN.2.AHL.TZ0.7b.ii:
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Find an expression for the acceleration of the ball at time $t$ .
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Find the value of $t$ at the instant the string breaks.
21N.1.AHL.TZ0.8a.i:
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21N.1.AHL.TZ0.8a.ii:
Interpret this point in the given context.

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Topic 5—Calculus