Introduction to integration — Question 170

Original QCE Vault practice · 5 marks

Q170 · Practice questionTechnology-freeComplex unfamiliar5 marks

QUESTION 170 (5 marks)

A particle moves on a straight line with acceleration

a(t)=6e^{-t}

m/s

^2

, initial velocity

v(0)=-1

m/s and initial position

s(0)=2

m. Determine the first time after

t=0

at which the particle is at rest, and determine its position at that time.

WORKED SOLUTION

5 marks

ANSWER

t=\ln(6/5)\approx0.182

s and

s=1+5\ln(6/5)\approx1.912

Worked solution

Integrating acceleration gives

v(t)=5-6e^{-t}

. Setting

v=0

gives

e^{-t}=5/6

, so

t=\ln(6/5)

. Integrating velocity gives

s(t)=5t+6e^{-t}-4

after using

s(0)=2

. Substitute the stopping time and

e^{-t}=5/6

to obtain

s=1+5\ln(6/5)

Integrates acceleration to obtain velocity.

[1 mark]

Uses the initial velocity correctly.

[1 mark]

Solves

v(t)=0

[1 mark]

Integrates velocity and applies the initial position.

[1 mark]

Evaluates the position at the stopping time.

[1 mark]

Practice question aligned to the current QCAA syllabus; review the worked solution and marking criteria.

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