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Date May 2014 Marks available 3 Reference code 14M.2.sl.TZ2.9
Level SL only Paper 2 Time zone TZ2
Command term Show that Question number 9 Adapted from N/A

Question

A particle moves in a straight line. Its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\), at time \(t\) seconds, is given by

\[v = {\left( {{t^2} - 4} \right)^3},{\text{ for }}0 \leqslant t \leqslant 3.\]

 

Find the velocity of the particle when \(t = 1\).

[2]
a.

Find the value of \(t\) for which the particle is at rest.

[3]
b.

Find the total distance the particle travels during the first three seconds.

[3]
c.

Show that the acceleration of the particle is given by \(a = 6t{({t^2} - 4)^2}\).

[3]
d.

Find all possible values of \(t\) for which the velocity and acceleration are both positive or both negative.

[4]
e.

Markscheme

substituting \(t = 1\) into \(v\)     (M1)

eg     \(v(1),{\text{ }}{\left( {{1^2} - 4} \right)^3}\)

velocity \( =  - 27{\text{ }}\left( {{\text{m}}{{\text{s}}^{ - 1}}} \right)\)     A1     N2

[2 marks]

a.

valid reasoning     (R1)

eg     \(v = 0,{\text{ }}{\left( {{t^2} - 4} \right)^3} = 0\)

correct working     (A1)

eg     \({t^2} - 4 = 0,{\text{ }}t =  \pm 2\), sketch

\(t = 2\)     A1     N2

[3 marks]

b.

correct integral expression for distance     (A1)

eg     \(\int_0^3 {\left| v \right|,{\text{ }}\int {\left| {{{\left( {{t^2} - 4} \right)}^3}} \right|,{\text{ }} - \int_0^2 {v{\text{d}}t + \int_2^3 {v{\text{d}}t} } } } \),

\(\int_0^2 {{{\left( {4 - {t^2}} \right)}^3}{\text{d}}t + \int_2^3 {{{\left( {{t^2} - 4} \right)}^3}{\text{d}}t} }\) (do not accept \(\int_0^3 {v{\text{d}}t} \))

\(86.2571\)

\({\text{distance}} = 86.3{\text{ (m)}}\)     A2     N3

[3 marks]

c.

evidence of differentiating velocity     (M1)

eg     \(v'(t)\)

\(a = 3{\left( {{t^2} - 4} \right)^2}(2t)\)     A2

\(a = 6t{\left( {{t^2} - 4} \right)^2}\)     AG     N0

[3 marks]

d.

METHOD 1

valid approach     M1

eg     graphs of \(v\) and \(a\)

correct working     (A1)

eg     areas of same sign indicated on graph

\(2 < t \leqslant 3\)   (accept \(t > 2\))     A2     N2

METHOD 2

recognizing that \(a \geqslant 0\) (accept \(a\) is always positive) (seen anywhere)     R1

recognizing that \(v\) is positive when \(t > 2\) (seen anywhere)     (R1)

\(2 < t \leqslant 3\)   (accept \(t > 2\))     A2     N2

[4 marks]

e.

Examiners report

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

Syllabus sections

Topic 6 - Calculus » 6.6 » Kinematic problems involving displacement \(s\), velocity \(v\) and acceleration \(a\).
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