2022 HCI P1 Q12

Timothy Gan

2022 HCI P1 Q12

A cargo drone is used to unload First Aid kit at an accident location in a remote mountain area. The First Aid kit is unloaded vertically through air. The speed of the kit, $v$ms$^{-1}$, is the rate of change of distance,$x$ m, of the kit measured vertically away from the drone with respect to time, $t$ seconds.

(i)

Write down a differential equation relating $v$, $x$ and $t$.

[1]

The motion of the First Aid kit is modelled by the differential equation

$\frac{{{\text{d}}^{2}}x}{\text{d}{{t}^{2}}}+\alpha {{\left( \frac{\text{d}x}{\text{d}t} \right)}^{2}}=10$, $\left( \text{A} \right)$

where $\alpha $ is a constant.

(ii)

By using the result in part (i), show that the differential equation $\left( \text{A} \right)$ can be expressed as

$\frac{\text{d}v}{\text{d}t}=10-\alpha {{v}^{2}}$

[1]

It is given that when $t=0$, $v=0$. It is also given that $\frac{\text{d}v}{\text{d}t}=4.375$ when $v=1.5$.

(iii)

Find the value of $\alpha $. By solving the differential equation in part (ii), show that

$v=\frac{k-k{{\text{e}}^{-10t}}}{m+{{\text{e}}^{-10t}}}$,

where $k$ and $m$ are constants to be determined.

[4]

(iv)

Sketch the graph of $v$ against $t$ and describe the behaviour of $\frac{\text{d}x}{\text{d}t}$ and $\frac{{{\text{d}}^{2}}x}{\text{d}{{t}^{2}}}$ in the long run.

[4]

(v)

Show that the area under the graph in part (iv) bounded by the lines $t=0$ and $t=T$ can be expressed as $\frac{2}{5}\ln \left( \frac{{{\text{e}}^{\beta \,T}}+{{\text{e}}^{-\beta \,T}}}{2} \right)$, where $\beta $ is a positive integer to be determined.

[3]

(vi)

What can be said about $\frac{2}{5}\ln \left( \frac{{{\text{e}}^{\beta \,T}}+{{\text{e}}^{-\beta \,T}}}{2} \right)$ in the context of this question?

[1]

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Published: 30th January 2023

Written by

Timothy Gan

This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.

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