A particle moves along a straight line. Its velocity, \(v(t) = t^3 – 8t^2 + 15t\) m s\(^{-1}\), at time \(t\) seconds is given for \(t \ge 0\).
(a) Find the times when the particle is instantaneously at rest. [3]
(b) Find the acceleration of the particle when \(t = 4\) seconds. [3]
(c) Find the displacement of the particle from its initial position after the first 6 seconds. [3]
(d) Find the total distance travelled by the particle in the first 6 seconds. [5]
Question
Find the times when the particle is instantaneously at rest. [3]
Hint
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Factor the velocity function \(v(t) = t^3 – 8t^2 + 15t\) and solve \(v(t)=0\).
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\(t = 0, 3, 5\) seconds
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Question
Find the acceleration of the particle when \(t = 4\) seconds. [3]
Hint
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Hint
Differentiate \(v(t)= t^3 – 8t^2 + 15t\) to find \(a(t)\) and substitute \(t = 4\).
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\(a(4) = -1 \; m\,s^{-2}\)
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Question
Find the displacement of the particle from its initial position after the first 6 seconds. [3]
Hint
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Hint
Integrate the velocity function \(v(t)\) from \(t=0\) to \(t=6\).
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\(s = -18 \; m\)
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Question
Find the total distance travelled by the particle in the first 6 seconds. [5]
Hint
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Determine where \(v(t)\) changes sign (using the zeros) and integrate the absolute value of \(v(t)\) over each interval.
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\(24 \; m\)
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