Assume that temperature varies linearly with height near the Earth's surface. Considering temperature at the surface of the Earth `T_1` and `T_2` at a height h above the surface, calculate the time t needed for a sound wave produced at a height. x. to reach the Earth's surface. Velocity of sound at the Earth's surface is c.

Options

A. `t = (2h)/(c ) (sqrtT_1)/((T_2 - T_1)) [ sqrtT_1 - sqrt((T_2 - T_1)/(h))x + T_1]`
B. `t = (2h)/(c ) (sqrtT_1)/((T_1 - T_2)) [sqrtT_1 - sqrt((T_2 - T_1)/(h))x + T_1]`
C. `t = (3h)/( c) (sqrtT_1)/((T_2 - T_1)) [ sqrtT_1 - sqrt((T_2-T_1)/(h)) x+ T_1]`
D. `t = (6h)/(c )(sqrtT_1)/((T__2- T_1)) [ sqrtT_1 - sqrt((T_2 - T_1)/(h)) x + T_1 ]`

Answer

To solve the problem of calculating the time \( t \) needed for a sound wave produced at a height \( x \) to reach the Earth's surface, we will follow these steps:

### Step 1: Understand the relationship between temperature and height
Given that temperature varies linearly with height, we can express the temperature at height \( h \) as:
\[ T(h) = T_1 + \left(\frac{T_2 - T_1}{h}\right) \cdot z \]
where \( z \) is the height above the surface, ranging from \( 0 \) to \( h \).

### Step 2: Determine the speed of sound at height \( z \)
The speed of sound \( v(z) \) in a gas is given by the formula:
\[ v(z) = \sqrt{\frac{\gamma P(z)}{\rho(z)}} \]
where \( P(z) \) is the pressure at height \( z \), \( \rho(z) \) is the density at height \( z \), and \( \gamma \) is the adiabatic constant.

However, for our case, we can simplify this by using the relationship of speed of sound with temperature:
\[ v(z) = k \sqrt{T(z)} \]
where \( k \) is a constant that depends on the gas properties.

### Step 3: Calculate the speed of sound at height \( x \)
At height \( x \), the temperature is:
\[ T(x) = T_1 + \left(\frac{T_2 - T_1}{h}\right) \cdot x \]
Thus, the speed of sound at height \( x \) is:
\[ v(x) = k \sqrt{T(x)} = k \sqrt{T_1 + \left(\frac{T_2 - T_1}{h}\right) \cdot x} \]

### Step 4: Calculate the time \( t \) for sound to travel from height \( x \) to the surface
The time taken \( t \) for the sound to travel a distance \( (h - x) \) is given by:
\[ t = \frac{\text{Distance}}{\text{Speed}} = \frac{h - x}{v(x)} \]
Substituting \( v(x) \):
\[ t = \frac{h - x}{k \sqrt{T_1 + \left(\frac{T_2 - T_1}{h}\right) \cdot x}} \]

### Step 5: Final expression for time \( t \)
If we consider the speed of sound at the surface \( c \) (which corresponds to \( T_1 \)), we can express \( k \) in terms of \( c \):
\[ c = k \sqrt{T_1} \]
Thus, we can rewrite the time \( t \) in terms of \( c \):
\[ t = \frac{(h - x) \sqrt{T_1}}{c \sqrt{T_1 + \left(\frac{T_2 - T_1}{h}\right) \cdot x}} \]

This gives us the time \( t \) needed for the sound wave to reach the Earth's surface.