The rate of decrease of atmospheric pressure $P$ Pascals (Pa) with respect to altitude $h$ kilom

Question

The rate of decrease of atmospheric pressure $P$ Pascals (Pa) with respect to altitude $h$ kilometres above sea level, is proportional to $\frac{P}{T}$, where $T$ is the temperature measured in Kelvin (K). The temperature $T$ decreases linearly by $6.5$ K for every one-kilometre increase in the altitude above sea level. It is given that the temperature is $293$ K at sea level.

Find a differential equation for $\frac{\mathrm{d}P}{\mathrm{d}h}$ in terms of $P$ and $h$.[2]
Given that 𝑃 >0, solve this model in part (i) to show that
$P=A|293-6.5h|^b,$ where 𝐴 and 𝑏 are constants.
Explain whether 𝑏 is positive or negative.
[4]

It is given that the atmospheric pressures are $101 \, 300$ Pa at sea level and $80 \, 000$ Pa at $h=2$.

Find the atmospheric pressure at the top of Mount Everest with an altitude of $8848$ metres.[4]
For $h\ge 0$, sketch the graph of $P$ against $h$. Explain a limitation of this model.[3]

Accepted Answer

(i) $\frac{\mathrm dP}{\mathrm dh}=-\frac{kP}{293-6.5h}$, $k>0$. (ii) $P=A|293-6.5h|^b$, $A>0$, $b=\frac{k}{6.5}>0$. (iii) $32500$ Pa. (iv) Graph shown; physical model requires $0\le h<\frac{586}{13}$ km.

2025 NJC P2 Q4

2025 NJC P2 Q4

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