Graphs 9
4 marks · HardA particle moves along a straight line such that its displacement after t seconds is given by , where s is in metres.
Find the times when the particle is at rest.
Explain how you would determine whether the particle is speeding up or slowing down at seconds.
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(a) t = 2, t = 3 seconds
(b) Compare the signs of velocity and acceleration at t = 2; if they are the same the particle is speeding up, if different, slowing down.
Common mistakes: Not differentiating s for velocity; Not setting velocity to zero; Incorrect reasoning about signs
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Finds expression for velocity and sets it to zero.
A1 (a) · 1 mark · Solves correctly for t and gives all positive solutions.
M2 (b) · 1 mark · Gives correct reasoning involving signs of velocity and acceleration.
A2 (b) · 1 mark · Explains that same signs means speeding up, different means slowing down. (follow-through)
Worked solution
Velocity:
Set :
So, t = 2 and t = 3 seconds.
At t = 2, compute velocity and acceleration:
At t = 2, v = 0. To determine if speeding up or slowing down, check velocity just before/after and sign of acceleration.
If v and a have the same sign around t = 2, the particle is speeding up; if different, slowing down.