Consider a car moving on a straight road with a speed of `100m//s`. The distance at which car can be stopped is `[mu_k=0.5]`

Options

A. 100 m
B. 1000 m
C. 800 m
D. 400 m

Answer

To solve the problem of determining the stopping distance of a car moving at a speed of \(100 \, \text{m/s}\) with a coefficient of kinetic friction \(\mu_k = 0.5\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Car**:
- The car experiences a gravitational force downward, which is \(mg\) (where \(m\) is the mass of the car and \(g\) is the acceleration due to gravity, approximately \(10 \, \text{m/s}^2\)).
- The normal force \(N\) acting upward is equal to the gravitational force when the car is on a flat surface, so \(N = mg\).

2. **Calculate the Kinetic Friction Force**:
- The kinetic friction force \(F_k\) can be calculated using the formula:
\[ F_k = \mu_k \cdot N = \mu_k \cdot mg \]
- Substituting \(\mu_k = 0.5\):
\[ F_k = 0.5 \cdot mg \]

3. **Apply the Work-Energy Principle**:
- The work done by the friction force is equal to the change in kinetic energy of the car. The initial kinetic energy \(K_i\) when the car is moving is given by:
\[ K_i = \frac{1}{2} mv^2 \]
- The final kinetic energy \(K_f\) when the car stops is \(0\). Therefore, the work done by friction can be expressed as:
\[ W = K_f - K_i = 0 - \frac{1}{2} mv^2 = -\frac{1}{2} mv^2 \]

4. **Set Up the Equation**:
- The work done by the friction force over the stopping distance \(s\) is:
\[ W = -F_k \cdot s = -\left(0.5 \cdot mg\right) \cdot s \]
- Equating the work done by friction to the change in kinetic energy:
\[ -0.5 \cdot mg \cdot s = -\frac{1}{2} mv^2 \]

5. **Simplify the Equation**:
- Cancel out the mass \(m\) from both sides (assuming \(m \neq 0\)):
\[ 0.5 g s = \frac{1}{2} v^2 \]
- Rearranging gives:
\[ s = \frac{v^2}{g} \]

6. **Substitute the Values**:
- Given \(v = 100 \, \text{m/s}\) and \(g = 10 \, \text{m/s}^2\):
\[ s = \frac{(100)^2}{10} = \frac{10000}{10} = 1000 \, \text{m} \]

### Final Answer:
The distance at which the car can be stopped is \(1000 \, \text{m}\).