Mathematical Methods — Question 12

QCAA 2020, Paper 1 · 5 marks

Q12 · 2020 · Technology-freeSimple familiar5 marks

QUESTION 12 (5 marks)

An object is moving in a straight line from a fixed point. The object is at the origin initially. The acceleration

a

(in

\mathrm{m\,s^{-2}}

) of the object is given by

a(t)=\pi\cos(\pi t),\quad t\ge0,

where

t

is time in seconds. The velocity at

t=1

0.5\;\mathrm{m\,s^{-1}}

Determine the initial acceleration.

[1 mark]

Determine the initial velocity.

[2 marks]

Determine the displacement after one second.

[2 marks]

WORKED SOLUTION

5 marks

ANSWER

\pi\;\mathrm{m\,s^{-2}}

; b)

0.5\;\mathrm{m\,s^{-1}}

; c)

0.5+\frac{2}{\pi}\;\mathrm m

Worked solution

a(0)=\pi\cos(0)=\pi\;\mathrm{m\,s^{-2}}

v(t)=\int a(t)dt=\sin(\pi t)+c.

Given

v(1)=0.5

c=0.5

, so

v(0)=0.5\;\mathrm{m\,s^{-1}}

s(t)=-\frac1\pi\cos(\pi t)+0.5t+c.

Given

s(0)=0

c=1/\pi

. Hence

s(1)=0.5+\frac{2}{\pi}\;\mathrm m.

correctly determines the initial acceleration

[1 mark]

correctly determines the general function

v(t)

[1 mark]

determines initial velocity

[1 mark]

determines general function

s(t)

[1 mark]

determines displacement after one second

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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