If x is the distance at which a car can be stopped when initially it was moving with speed u, then on making the speed of the car Nu, the distance at which the car can be stopped is

Options

A. Nu
B. `N^2 x`
C. `(u)/(N )`
D. `(u )/(N^2)`

Answer

To solve the problem, we need to determine how the stopping distance of a car changes when its speed is increased. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Initial Condition**:
- The car is initially moving with speed \( u \) and can stop in a distance \( x \).

2. **Apply the Kinematic Equation**:
- We can use the kinematic equation that relates initial velocity, final velocity, acceleration, and distance:
\[ v^2 = u^2 + 2as \]
- Here, \( v \) is the final velocity (which is 0 when the car stops), \( u \) is the initial velocity, \( a \) is the acceleration (which will be negative since it is deceleration), and \( s \) is the distance traveled (which is \( x \) in this case).

3. **Set Up the Equation for Initial Speed \( u \)**:
- Plugging in the values for the initial condition:
\[ 0 = u^2 + 2(-a)x \]
- Rearranging gives:
\[ u^2 = 2ax \]
- Thus, the acceleration \( a \) can be expressed as:
\[ a = \frac{u^2}{2x} \]

4. **Consider the New Speed \( Nu \)**:
- Now, if the speed of the car is increased to \( Nu \), we need to find the new stopping distance \( x' \).
- Again applying the kinematic equation:
\[ 0 = (Nu)^2 + 2(-a)x' \]
- This simplifies to:
\[ 0 = N^2u^2 + 2(-a)x' \]
- Rearranging gives:
\[ N^2u^2 = 2ax' \]

5. **Substituting the Expression for Acceleration**:
- We know from the previous step that \( a = \frac{u^2}{2x} \). Substituting this into the equation gives:
\[ N^2u^2 = 2\left(\frac{u^2}{2x}\right)x' \]
- This simplifies to:
\[ N^2u^2 = \frac{u^2}{x}x' \]

6. **Solving for \( x' \)**:
- Dividing both sides by \( u^2 \) (assuming \( u \neq 0 \)):
\[ N^2 = \frac{x'}{x} \]
- Rearranging gives:
\[ x' = N^2x \]

### Conclusion:
The distance at which the car can be stopped when its speed is increased to \( Nu \) is \( N^2x \).