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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