Consider a distance-time polynomial function \(s(t) = 3t^3 – 5t^2 + 2t + 7\) representing the position of a vehicle along a straight road, where time \(t\) is in seconds and \(s(t)\) is in meters.
Answer the following:
(a) For SL students: Differentiate \(s(t)\) with respect to \(t\) to find the velocity function \(v(t)=s'(t)\).
(b) For HL students: Differentiate the velocity function \(v(t)\) to obtain the acceleration function \(a(t)=v'(t)\).
(c) Interpret the motion of the vehicle at \(t = 2\) seconds by evaluating \(v(2)\) and \(a(2)\) and explaining what these values indicate about the vehicle’s motion.
Question
For SL level: Differentiate \(s(t) = 3t^3 – 5t^2 + 2t + 7\) with respect to \(t\) to obtain the velocity function \(v(t)\).
Hint
Hint
Hint
Recall that the derivative of \(t^n\) is \(n t^{n-1}\).
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- Formula Booklet
Complete Answer
\(v(t) = 9t^2 – 10t + 2\)
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Question
For HL level: Differentiate the velocity function \(v(t) = 9t^2 – 10t + 2\) with respect to \(t\) to obtain the acceleration function \(a(t)\).
Hint
Hint
Hint
Differentiate each term of \(v(t)\) term by term.
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- Formula Booklet
Complete Answer
\(a(t) = 18t – 10\)
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Question
Interpret the motion of the vehicle at \(t = 2\) seconds by evaluating \(v(2)\) and \(a(2)\), and explain what these values indicate about the vehicle’s motion.
Hint
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Hint
Substitute \(t=2\) into the expressions for \(v(t)\) and \(a(t)\) and consider the signs of the results.
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- Formula Booklet
Complete Answer
At \(t = 2\), \(v(2) = 9(2)^2 – 10(2) + 2 = 18\) m/s and \(a(2) = 18(2) – 10 = 26\) m/s\(^2\). This indicates that the vehicle is moving forward with a positive velocity and is accelerating, meaning it is speeding up.
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