An automobile travelling at 50 km `h^(-1)` ,can be stopped at a distance of 40 cm by applying brakes. If the same automobile is travelling at `90 km h^(-1)` ,all others conditions remaining the same and assuming no skidding, the minimum stopping distance in cm is

Options

A. (a)72
B. (b)92.5
C. (c)102.6
D. (d)129.6

Answer

To solve the problem of finding the minimum stopping distance of an automobile when traveling at different speeds, we can use the relationship between stopping distance, speed, and acceleration. The stopping distance \( S \) is given by the formula:

\[ S = \frac{V^2}{2a} \]

Where:
- \( S \) is the stopping distance,
- \( V \) is the velocity,
- \( a \) is the deceleration (which remains constant in this case).

### Step 1: Identify the known values for the first scenario
For the first scenario:
- Speed \( V_1 = 50 \) km/h
- Stopping distance \( S_1 = 40 \) cm

### Step 2: Convert speed from km/h to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):

\[ V_1 = 50 \times \frac{5}{18} = \frac{250}{18} \approx 13.89 \text{ m/s} \]

### Step 3: Use the stopping distance formula for the first scenario
Using the stopping distance formula:

\[ S_1 = \frac{V_1^2}{2a} \]

Substituting the known values:

\[ 40 = \frac{(13.89)^2}{2a} \]

### Step 4: Solve for acceleration \( a \)
Rearranging the equation to find \( a \):

\[ 2a \cdot 40 = (13.89)^2 \]

\[ 80a = 193.61 \quad \Rightarrow \quad a = \frac{193.61}{80} \approx 2.42 \text{ m/s}^2 \]

### Step 5: Identify the known values for the second scenario
For the second scenario:
- Speed \( V_2 = 90 \) km/h

### Step 6: Convert speed from km/h to m/s for the second scenario
Using the same conversion factor:

\[ V_2 = 90 \times \frac{5}{18} = \frac{450}{18} = 25 \text{ m/s} \]

### Step 7: Use the stopping distance formula for the second scenario
Using the stopping distance formula again:

\[ S_2 = \frac{V_2^2}{2a} \]

Substituting the known values:

\[ S_2 = \frac{(25)^2}{2 \times 2.42} \]

### Step 8: Calculate \( S_2 \)
Calculating \( S_2 \):

\[ S_2 = \frac{625}{4.84} \approx 129.6 \text{ m} \]

### Step 9: Convert \( S_2 \) from meters to centimeters
Since we need the answer in centimeters:

\[ S_2 = 129.6 \times 100 = 12960 \text{ cm} \]

### Final Answer
Thus, the minimum stopping distance when the automobile is traveling at 90 km/h is approximately:

\[ S_2 \approx 129.6 \text{ cm} \]