Question 8
The position vectors of two objects over time, $t$, where $t\ge0$, are given by
$$\begin{aligned}\mathbf{r}_1(t)&=-2\,\hat{\mathbf{i}}+t^2\,\hat{\mathbf{j}}\\\mathbf{r}_2(t)&=at\,\hat{\mathbf{i}}+4\,\hat{\mathbf{j}}\qquad\text{where }(a\in\mathbb{R})\end{aligned}$$
Given that the two objects collide, the value of $a$ is
That’s it. Step through the hints below to see why. In 2025, 83.3% of Queensland students chose (C).
Not quite. With $a=2$, object 2 moves to the right, away from $x=-2$. It can never reach object 1. In 2025, 3.9% of Queensland students chose (A).
Not quite. With $a=1$, object 2 moves to the right, away from $x=-2$. It can never reach object 1. In 2025, 5.9% of Queensland students chose (B).
Not quite. With $a=-2$, the paths cross at $(-2,4)$, but object 2 gets there at $t=1$ and object 1 at $t=2$. Crossing paths isn’t a collision. In 2025, 6.7% of Queensland students chose (D).
Both objects are at $(-2,4)$ at the same time. They collide.
Object 2 is at $(-2,4)$ now, but object 1 is still below it. Object 1 only gets there at $t=2$, after object 2 has gone.
Object 2 moves right, away from the line $x=-2$, so the objects never meet.
The objects are in different places at this time. Move the slider to see whether they ever meet.
Two objects collide when they are at the same place at the same time. Set $\mathbf{r}_1(t)=\mathbf{r}_2(t)$ and equate the $\hat{\mathbf{i}}$ and $\hat{\mathbf{j}}$ components separately, using the same $t$ in both.
$$\hat{\mathbf{i}}:\ -2=at\qquad\hat{\mathbf{j}}:\ t^2=4$$
Start with the equation that has only $t$ in it.
$$t^2=4\quad\Rightarrow\quad t=2$$
$t=-2$ is rejected because the question says $t\ge0$.
Substitute $t=2$ into the other equation.
$$-2=2a\quad\Rightarrow\quad a=-1$$
Try $a=-2$ in the diagram as well. The paths still cross at $(-2,4)$, but the objects get there at different times.
$$-2\,\hat{\mathbf{i}}+t^2\,\hat{\mathbf{j}}=at\,\hat{\mathbf{i}}+4\,\hat{\mathbf{j}}$$
$$\hat{\mathbf{j}}:\ t^2=4\ \Rightarrow\ t=2\ (t\ge0)\qquad\hat{\mathbf{i}}:\ -2=2a\ \Rightarrow\ a=-1$$
The objects collide at $(-2,4)$ when $t=2$.
How Queensland students answered in 2025
From the QCAA Specialist Mathematics subject report for 2025. The percentages can add to less than 100 because some students didn’t answer.
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Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA Specialist Mathematics external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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