A particle moves in a straight line. The displacement of the particle at time t s is s m, where
\(s = t^3 - 6t^2 + 4t\).
Find the velocity of the particle at the instant when its acceleration is zero.
Solution
First, find the velocity \(v\) by differentiating the displacement \(s\) with respect to time \(t\):
\(v = \frac{ds}{dt} = 3t^2 - 12t + 4\).
Next, find the acceleration \(a\) by differentiating the velocity \(v\) with respect to time \(t\):
\(a = \frac{dv}{dt} = 6t - 12\).
Set the acceleration to zero to find the time \(t\) when the acceleration is zero:
\(6t - 12 = 0\)
\(t = 2\).
Substitute \(t = 2\) into the velocity equation to find the velocity at this time:
\(v = 3(2)^2 - 12(2) + 4 = 12 - 24 + 4 = -8 \text{ ms}^{-1}\).
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