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Date November 2019 Marks available 4 Reference code 19N.2.SL.TZ0.S_10
Level Standard Level Paper Paper 2 Time zone Time zone 0
Command term Find Question number S_10 Adapted from N/A

Question

A rocket is travelling in a straight line, with an initial velocity of 140 m s−1. It accelerates to a new velocity of 500 m s−1 in two stages.

During the first stage its acceleration, a m s−2, after t seconds is given by a(t)=240sin(2t), where 0tk.

The first stage continues for k seconds until the velocity of the rocket reaches 375 m s−1.

Find an expression for the velocity, v m s−1, of the rocket during the first stage.

[4]
a.

Find the distance that the rocket travels during the first stage.

[4]
b.

During the second stage, the rocket accelerates at a constant rate. The distance which the rocket travels during the second stage is the same as the distance it travels during the first stage.

Find the total time taken for the two stages.

[6]
c.

Markscheme

recognizing that v=a        (M1)

correct integration         A1

eg      120cos(2t)+c

attempt to find c using their v(t)        (M1)

eg      120cos(0)+c=140

v(t)=120cos(2t)+260         A1   N3

[4 marks]

a.

evidence of valid approach to find time taken in first stage           (M1)

eg      graph,  120cos(2t)+260=375

k=1.42595         A1

attempt to substitute their v and/or their limits into distance formula           (M1)

eg      1.425950|v|,   260120cos(2t),   k0(260120cos(2t))dt

353.608

distance is 354 (m)         A1   N3

[4 marks]

b.

recognizing velocity of second stage is linear (seen anywhere)          R1

eg      graph,   s=12h(a+b),   v=mt+c

valid approach           (M1)

eg      v=353.608

correct equation           (A1)

eg      12h(375+500)=353.608

time for stage two =0.808248  (0.809142 from 3 sf)         A2

2.23420  (2.23914 from 3 sf)

2.23 seconds  (2.24 from 3 sf)         A1   N3

[6 marks]

c.

Examiners report

[N/A]
a.
[N/A]
b.
[N/A]
c.

Syllabus sections

Topic 3— Geometry and trigonometry » SL 3.8—Solving trig equations
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Topic 2—Functions
Topic 3— Geometry and trigonometry
Topic 5 —Calculus

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