Question 10
An object travels in a straight line so that its velocity at time $t$ seconds is given by $v(t)=2t+\sin(2t)$. Determine the expression for acceleration as a function of time.
That’s it. Step through the hints below to see why. In 2021, 82.7% of Queensland students chose (A).
Not quite. The 2 is right, but $-\dfrac{1}{2}\cos(2t)$ is what you get by integrating $\sin(2t)$. Differentiating $\sin(2t)$ gives $2\cos(2t)$ by the chain rule. In 2021, 4.3% of Queensland students chose (B).
Not quite. $2\cos(2t)$ is right, but $t^2$ comes from integrating $2t$. Differentiating $2t$ gives 2. In 2021, 4.6% of Queensland students chose (C).
Not quite. $t^2-\dfrac{1}{2}\cos(2t)$ is an antiderivative of $v(t)$. Integrating velocity leads to displacement. Acceleration is the derivative of velocity, so differentiate instead. In 2021, 7.9% of Queensland students chose (D).
Option (A) gives the gradient of the velocity graph at every time, so the dashed blue line lies on the orange tangent. Acceleration is the derivative of velocity.
Here option () gives , the same as the tangent’s gradient. That is a coincidence at this time. Move the point and the lines split.
Very close, but not equal. Option () gives here and the tangent’s gradient is . Move the point and the lines split.
Option () gives here, but the tangent’s gradient is . The dashed blue line is not the tangent.
Acceleration is the rate of change of velocity. So differentiate $v(t)$. Don’t integrate: that takes you back to displacement.
$$a(t)=v'(t)$$
On the graph, the acceleration at any time is the gradient of the tangent to the velocity curve.
Differentiate each term. For $\sin(2t)$, the chain rule multiplies by the derivative of $2t$, which is 2.
$\displaystyle \frac{d}{dt}(2t)=2$
$\displaystyle \frac{d}{dt}\sin(2t)=2\cos(2t)$
$$a(t)=2+2\cos(2t)$$
The other options come from integrating. $t^2$ is the antiderivative of $2t$, and $-\tfrac12\cos(2t)$ is the antiderivative of $\sin(2t)$.
$\displaystyle a(t)=v'(t)$$\displaystyle =\frac{d}{dt}\bigl(2t+\sin(2t)\bigr)$
$\displaystyle a(t)=2+2\cos(2t)$
Differentiate each term, using the chain rule for $\sin(2t)$.
How Queensland students answered in 2021
From the QCAA Mathematical Methods subject report for 2021. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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