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

Question

A particle P moves along a straight line. Its velocity v P  m s 1 after t seconds is given by v P = t sin ( π 2 t ) , for 0 t 8 . The following diagram shows the graph of v 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 t 8 .

[2]
a.ii.

A second particle Q also moves along a straight line. Its velocity, v Q  m s 1 after t seconds is given by v Q = t for 0 t 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 0 8 | v | d t ,   | v Q | d t ,   0 2 v d t 2 4 v d t + 4 6 v d t 6 8 v d t

9.64782

distance = 9.65  (metres)     A1     N2

[2 marks]

a.ii.

correct approach     (A1)

eg s = t ,   0 k t d t ,   0 k | v Q | d t

correct integration     (A1)

eg t = 2 3 t 3 2 + c ,   [ 2 3 x 3 2 ] 0 k ,   2 3 k 3 2

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

eg 2 3 k 3 2 = 9.65 ,   0 k t 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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