(

a

) Show that Eq. 15-13, the variation of pressure with alti-

tude in the atmosphere (temperature assumed to be uniform),

can be written in terms of density



as

(

a

) Show that Eq. 15-13, the variation of pressure with alti-

tude in the atmosphere (temperature assumed to be uniform),

can be written in terms of density



as

where



0

is the density at the ground (

h



0). (

b

) Assume that

the drag force

D

due to the air on an object moving at speed

v

is given by

D



CA



v

2

where

C

is a constant,

A

is the frontal

cross-sectional area of the object, and



is the local air den-

sity. Find the altitude at which the drag force on a rocket is a

maximum if the rocket is launched vertically and moves with

constant upward acceleration

a

r

.(

a

) Show that Eq. 15-13, the variation of pressure with alti-

tude in the atmosphere (temperature assumed to be uniform),

can be written in terms of density



as

where



0

is the density at the ground (

h



0). (

b

) Assume that

the drag force

D

due to the air on an object moving at speed

v

is given by

D



CA



v

2

where

C

is a constant,

A

is the frontal

cross-sectional area of the object, and



is the local air den-

sity. Find the altitude at which the drag force on a rocket is a

maximum if the rocket is launched vertically and moves with

constant upward acceleration

a

r

.

To tackle this problem, we need to break it down into two parts. The first part involves deriving the relationship between pressure and density in the atmosphere, while the second part focuses on determining the altitude at which the drag force on a rocket is maximized. Let's dive into each part step by step.

Part (a): Pressure Variation with Altitude

We start with the hydrostatic equation, which describes how pressure changes with altitude in the atmosphere. The equation can be expressed as:

dP/dh = -ρg

Here, P is the pressure, h is the altitude, ρ is the density of the air, and g is the acceleration due to gravity. Assuming temperature is uniform, we can relate density to pressure using the ideal gas law:

P = ρRT

Where R is the specific gas constant and T is the absolute temperature. By substituting this expression for density into the hydrostatic equation, we can rewrite it as:

dP/dh = - (P/RT) g

Rearranging gives us:

dP/P = - (g/RT) dh

Integrating both sides, we find:

ln(P) = - (g/RT) h + C

Exponentiating both sides leads to:

P = P₀ e^(-gh/RT)

Where P₀ is the pressure at ground level. Now, to express this in terms of density, we can substitute back using the ideal gas law:

ρ = P/(RT)

Thus, we can express pressure in terms of density:

P = ρRT

Combining these equations, we can derive:

ρ = ρ₀ e^(-gh/RT)

Where ρ₀ is the density at ground level (h = 0). This shows how pressure varies with altitude in terms of density.

Part (b): Maximum Drag Force on a Rocket

Next, we need to analyze the drag force acting on a rocket. The drag force D is given by:

D = C A ρ v²

Where C is a constant, A is the frontal area, ρ is the local air density, and v is the speed of the rocket. As the rocket ascends, both the density of the air and the speed of the rocket change with altitude.

To find the altitude at which the drag force is maximized, we need to consider how density changes with altitude:

ρ(h) = ρ₀ e^(-gh/RT)

As the rocket moves upward with a constant acceleration aₗ, its speed increases, and we can express the speed as:

v = v₀ + aₗ t

Substituting this into the drag force equation gives:

D(h) = C A (ρ₀ e^(-gh/RT)) (v₀ + aₗ t)²

To find the maximum drag force, we need to differentiate this expression with respect to altitude h and set the derivative equal to zero:

dD/dh = 0

This will involve applying the product rule and chain rule to account for the exponential decay of density with altitude and the increasing speed of the rocket. The resulting equation will yield a critical point that can be solved for altitude h.

In practice, this calculation can become complex, but it essentially involves balancing the decrease in density with the increase in speed to find the altitude where the drag force reaches its maximum value.

Summary

In summary, we derived the relationship between pressure and density in the atmosphere, showing how pressure decreases with altitude under the assumption of uniform temperature. We also set up the framework to analyze the drag force on a rocket, leading to the determination of the altitude at which this force is maximized. This involves understanding the interplay between air density and the rocket's speed as it ascends.