# kinematics

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1)A particle moves in a straight line so that its distance, s metres, from a fixed point O is given by s = 2t^3 - 3t^2 - 12t + 6, where t is the time in seconds after passing O. Find the minimum speed attained by the particle

2)A particle moves in a straight line so that its distance, s meters, from a fixed point is given by s=t^3(2-t)^2, where t is the time in seconds after passing through the fixed point. What is the greatest distance travelled by the particle?

• kinematics -

Bailey, Andrea, Danny, Riley, anabelle, holly, LILLY, zachary, I'm stumped -- or whoever!

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• kinematics -

1) The speed at time t is
|ds/dt| = |6t^2 -6t -12|
(Speed is always positive)
The minimum value of |ds/dt| is 0 if there is a time when
6t^2 -6t -12 = 0
t^2 -t -2 = 0
(t-2)(t+1) = 0
So there IS a time when the speed is zero.

2) The greatest value of s occurs at a time where ds/dt = 0 is the second derivative d^2/dt^2 at that time is negative.

s grows without bound for very large t greater than 2. I do not see how there could be a greatest value of s unless you restrict the region of consideration to 0 < t <2
t = 2

• kinematics -

In my answer to 2), "is" should be "if".

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