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 $101300$ Pa at sea level and $80000$ 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)$\frac{dP}{dh}= \frac{kP}{293-6.5 h}$, k0 (iii)32500 Pa

2025 NJC P2 Q4

2025 NJC P2 Q4