Question 4
The displacement (m) of a moving particle is given by $d=e^{0.5t}-1$, where $t$ is time (s).
The acceleration $(\text{ms}^{-2})$ of the particle when $t=4$ is
Not quite. 7.3891 is $e^{2}=e^{0.5(4)}$. That’s what you get if you differentiate $e^{0.5t}$ without bringing down the 0.5. The chain rule multiplies by 0.5 each time, so $a=0.25e^{0.5t}$. In 2023, 3.3% of Queensland students chose (A).
Not quite. 6.3891 is $d(4)=e^{2}-1$, the displacement at $t=4$. Acceleration is the second derivative of displacement. In 2023, 6.6% of Queensland students chose (B).
Not quite. 3.6945 is $v(4)=0.5e^{2}$, the velocity. Differentiate once more to get the acceleration. In 2023, 8.5% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2023, 81.1% of Queensland students chose (D).
At $t=4$ each curve gives one of the options. The acceleration is the lowest, $a(4)=0.25e^{2}\approx1.8473$, option (D). The displacement gives (B), the velocity gives (C), and $e^{0.5t}$ gives (A).
At every time, $v$ is half of $e^{0.5t}$ and $a$ is half of $v$. Each derivative brings down another factor of 0.5.
Velocity is the derivative of displacement. The derivative of $e^{kt}$ is $ke^{kt}$, and the constant $-1$ disappears.
$$v(t)=d'(t)=0.5e^{0.5t}$$
Acceleration is the derivative of velocity. The chain rule brings down another 0.5.
$\displaystyle a(t)=v'(t)$$\displaystyle =0.5\times0.5e^{0.5t}$$\displaystyle =0.25e^{0.5t}$
$$a(4)=0.25e^{2}\approx1.8473$$
The acceleration at 4 seconds is about $1.8473\ \text{m s}^{-2}$. Check the order of the other options in the figure: $e^{2}$ is above $d(4)$, which is above $v(4)$, which is above $a(4)$.
$$\begin{aligned}v(t)&=0.5e^{0.5t}\\a(t)&=0.25e^{0.5t}\\a(4)&=0.25e^{2}\approx1.8473\end{aligned}$$
Differentiate twice, bringing down a factor of 0.5 each time, then substitute $t=4$.
How Queensland students answered in 2023
From the QCAA Mathematical Methods subject report for 2023. 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 2023 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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