Advanced Kinematics Integration Problem

00:00 7 marks

A particle moving along a straight line has an acceleration given by \(a(t) = 6t – 4\) m/s\(^2\), where \(t\) is time in seconds. Given that the initial velocity of the particle is \(v(0)=2\) m/s, answer the following:

(a) Determine the velocity function \(v(t)\) by integrating the acceleration function, making sure to use the initial condition to find the constant of integration.

(b) Using the velocity function obtained in part (a), evaluate the displacement of the particle over the time interval from \(t = 0\) to \(t = 3\) seconds.

Part a

4 minutes 3 marks

Question

Determine the velocity function \(v(t)\) for the particle. Integrate the acceleration \(a(t)=6t-4\) and use the initial condition \(v(0)=2\) to find the constant of integration.

Hint Hint

Hint

Integrate \(6t-4\) with respect to \(t\) and then substitute \(t=0\) to solve for the constant.

Answer Answer
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Complete Answer

v(t) = 3t^2 – 4t + 2

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Part b

6 minutes 4 marks

Question

Using the velocity function \(v(t) = 3t^2 – 4t + 2\) derived in part (a), evaluate the displacement of the particle over the interval \(t \in [0,3]\) seconds.

Hint Hint

Hint

Set up the definite integral of \(v(t)\) from \(t=0\) to \(t=3\) and evaluate.

Answer Answer
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Answer Answer

Complete Answer

15 m

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