Two cars of same mass are moving with velocities `v_(1) and v_(2)` respectively. If they are stopped by supplying same breaking power in time `t_(1) and t_(2)` respectively then `(v_(1))/(v_(2))` is

Options

A. `(t_(1))/(t_(2))`
B. `(t_(1)^(2))/(t_(2)^(2))`
C. `m(t_(1))/(t_(2))`
D. `sqrt((t_(1))/(t_(2)))`

Answer

To solve the problem, we need to derive the relationship between the velocities \( v_1 \) and \( v_2 \) of two cars that are stopped by the same braking power in different times \( t_1 \) and \( t_2 \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have two cars of the same mass moving with velocities \( v_1 \) and \( v_2 \) respectively. They are stopped by applying the same braking power in times \( t_1 \) and \( t_2 \).

2. **Braking Power Relation**:
The braking power \( P \) can be expressed as:
\[ P = F \cdot v \]
where \( F \) is the braking force and \( v \) is the velocity of the car. Since the force and velocity are in opposite directions, we can write:
\[ P_1 = -F_1 v_1 \quad \text{and} \quad P_2 = -F_2 v_2 \]
Given that the braking power is the same for both cars, we have:
\[ P_1 = P_2 \implies -F_1 v_1 = -F_2 v_2 \implies F_1 v_1 = F_2 v_2 \quad \text{(Equation 1)} \]

3. **Acceleration and Time Relation**:
Using the equation of motion, we know:
\[ v = u + at \]
For the first car, which stops in time \( t_1 \):
\[ 0 = v_1 - a_1 t_1 \implies v_1 = a_1 t_1 \quad \text{(Equation 2)} \]
For the second car, which stops in time \( t_2 \):
\[ 0 = v_2 - a_2 t_2 \implies v_2 = a_2 t_2 \quad \text{(Equation 3)} \]

4. **Relating Acceleration to Force**:
The acceleration can be expressed in terms of force and mass:
\[ a_1 = \frac{F_1}{m} \quad \text{and} \quad a_2 = \frac{F_2}{m} \]
Substituting these into Equations 2 and 3:
\[ v_1 = \frac{F_1}{m} t_1 \quad \text{and} \quad v_2 = \frac{F_2}{m} t_2 \]

5. **Finding the Ratio of Velocities**:
Now, we can find the ratio \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\frac{F_1}{m} t_1}{\frac{F_2}{m} t_2} = \frac{F_1 t_1}{F_2 t_2} \]
From Equation 1, we know \( F_1 v_1 = F_2 v_2 \) implies \( \frac{F_1}{F_2} = \frac{v_2}{v_1} \). Substituting this into the ratio gives:
\[ \frac{v_1}{v_2} = \frac{v_2}{v_1} \cdot \frac{t_1}{t_2} \]
Rearranging gives:
\[ \left(\frac{v_1}{v_2}\right)^2 = \frac{t_1}{t_2} \]
Therefore, we find:
\[ \frac{v_1}{v_2} = \sqrt{\frac{t_1}{t_2}} \]

### Final Answer:
\[ \frac{v_1}{v_2} = \sqrt{\frac{t_1}{t_2}} \]