Question

atmospheric pressure P varies inversely as the altitude h.​

Answer 1

Answer:

• The atmospheric pressure varies with the altitude $h$ as the density of air decreases because when we rise up, the air becomes rarer.

• Atmospheric pressure decreases with increases in altitude.

As we know that the density of air becomes heavier near the surface of the earth (due to gravity) and begins to lighten as we go to higher altitudes and eventually leads to empty space, that is outside the atmosphere of the earth.

Thus, the atmospheric pressure is high at lower altitudes, the density being higher. The atmospheric pressure is low at higher altitudes, the density being lower. So, atmospheric pressure varies inversely with the altitude $h$.

Answer 2

Given

Atmospheric pressure P

Altitude h

To prove

Atmospheric pressure p varies with altitude h

Proof

Atmospheric pressure p varies with altitude h accordant with the equation dP/ dh = kP.

The rate of change of pressure with reference to altitude is dP ⁄ dh = kP,

( rate of change varied directly with pressure)

where,

k is the constant of proportionality

P is the pressure

This is a differential accurate equation and it can be answered to get a general expression for the pressure with altitude.

Rearrange to group and integrate (as to solve as exponential equation)

∫ (1/ P) dP = k ∫ dh

limits of integration on P are P (h °) and P (h) and on h are h ° to h

ln P (h)-ln P (0)

= ln P (h) ⁄ P (0)

= k (h-0)

= kh

= ln P (h) ⁄ P (h °)

= k (h-h °)

Still,

, If the starting pressure is taken to be the pressure stumped position. Also h ° = 0 and P (h °) = P (0)

Take the exponential of each side of the equation results in

P (h) ⁄ P (0) = k

P (h) = P (0) k

This is an equation for the pressure in the terms of h, the altitude

and it's dependent on the original pressure.

Still, for this to be useful, we need to know the k.

Still, we will search for the value k as and when needed, If we replace the natural log type of the equation.