An automobile travelling with a speed `60 km//h` , can brake to stop within a distance of `20 m` . If the car is going twice as fast i. e. , `120 km//h`, the stopping distance will be

Options

A. 60 m
B. 40 m
C. 20 m
D. 80m

Explanation

To solve the problem, we will use the relationship between the initial speed of the automobile, the stopping distance, and the acceleration. We can apply the kinematic equation that relates these quantities.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Initial speed (u1) = 60 km/h
- Stopping distance (s1) = 20 m
- New speed (u2) = 120 km/h

2. **Convert Speeds from km/h to m/s:**
- To convert km/h to m/s, we use the conversion factor: \(1 \text{ km/h} = \frac{5}{18} \text{ m/s}\).
- Therefore,
- \(u1 = 60 \text{ km/h} = 60 \times \frac{5}{18} = \frac{300}{18} = 16.67 \text{ m/s}\)
- \(u2 = 120 \text{ km/h} = 120 \times \frac{5}{18} = \frac{600}{18} = 33.33 \text{ m/s}\)

3. **Use the Kinematic Equation:**
- The kinematic equation we will use is:
\[ v^2 = u^2 + 2as \]
- For stopping, the final velocity (v) is 0, so we can rearrange the equation to find acceleration (a):
\[ 0 = u^2 + 2as \implies a = -\frac{u^2}{2s} \]
- For the first case (u1 and s1):
\[ a = -\frac{(16.67)^2}{2 \times 20} \]

4. **Calculate the Acceleration for the First Case:**
- Calculate \( (16.67)^2 = 278.89 \)
- Therefore,
\[ a = -\frac{278.89}{40} = -6.97225 \text{ m/s}^2 \]

5. **Relate the Stopping Distances:**
- From the kinematic equation, we know that stopping distance is proportional to the square of the initial speed:
\[ \frac{s1}{s2} = \frac{u1^2}{u2^2} \]
- Rearranging gives:
\[ s2 = s1 \times \frac{u2^2}{u1^2} \]

6. **Substitute the Values:**
- We already have:
- \(u1 = 16.67 \text{ m/s}\)
- \(u2 = 33.33 \text{ m/s}\)
- \(s1 = 20 \text{ m}\)
- Now calculate \(s2\):
\[ s2 = 20 \times \frac{(33.33)^2}{(16.67)^2} \]
- Calculate \( (33.33)^2 = 1111.0889 \) and \( (16.67)^2 = 278.0889 \):
\[ s2 = 20 \times \frac{1111.0889}{278.0889} \approx 20 \times 4 = 80 \text{ m} \]

### Final Answer:
The stopping distance when the car is going at 120 km/h will be **80 meters**.