The root mean square speed of the molecule at constant pressure at temperature T is v, what is its root mean square speed, if temperature is reduced to `(T)/(2)`.

Explanation

To solve the problem, we need to understand the relationship between the root mean square speed of gas molecules and temperature. The root mean square speed (Vrms) of a gas is given by the formula:

\[ V_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]

where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 1: Identify the initial conditions
At the initial temperature \( T \), the root mean square speed is given as \( V \). Therefore, we can express this as:

\[ V = \sqrt{\frac{3RT}{M}} \]

### Step 2: Determine the new temperature
Now, the temperature is reduced to \( \frac{T}{2} \). We need to find the new root mean square speed \( V' \) at this new temperature.

### Step 3: Write the expression for the new root mean square speed
Using the same formula for root mean square speed at the new temperature \( \frac{T}{2} \):

\[ V' = \sqrt{\frac{3R\left(\frac{T}{2}\right)}{M}} \]

### Step 4: Simplify the expression
Now, simplify the expression for \( V' \):

\[ V' = \sqrt{\frac{3RT}{2M}} = \sqrt{\frac{3RT}{M} \cdot \frac{1}{2}} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{3RT}{M}} = \sqrt{\frac{1}{2}} \cdot V \]

### Step 5: Final expression
Thus, we can express \( V' \) in terms of \( V \):

\[ V' = \frac{V}{\sqrt{2}} \]

### Conclusion
The root mean square speed of the molecule when the temperature is reduced to \( \frac{T}{2} \) is:

\[ V' = \frac{V}{\sqrt{2}} \]