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. 72
B. 92.5
C. 102.6
D. 129.6

Answer

To solve the problem step by step, we will use the equations of motion to find the stopping distance of the automobile when traveling at different speeds.

### Step 1: Convert the speeds from km/h to m/s
We need to convert the speeds from kilometers per hour (km/h) to meters per second (m/s) using the conversion factor \( \frac{5}{18} \).

- For \( U_1 = 50 \, \text{km/h} \):
\[ U_1 = 50 \times \frac{5}{18} = \frac{250}{18} \approx 13.89 \, \text{m/s} \]

- For \( U_2 = 90 \, \text{km/h} \):
\[ U_2 = 90 \times \frac{5}{18} = \frac{450}{18} = 25 \, \text{m/s} \]

### Step 2: Use the equation of motion to find acceleration
We know from the problem that the stopping distance \( S_1 \) at speed \( U_1 \) is 40 cm (which is 0.4 m). We will use the third equation of motion:
\[ V^2 = U^2 + 2AS \]
where \( V = 0 \) (final velocity), \( U = U_1 \), \( A \) is the acceleration (deceleration in this case), and \( S = S_1 \).

Rearranging the equation to find \( A \):
\[ 0 = U_1^2 + 2AS_1 \implies A = -\frac{U_1^2}{2S_1} \]

Substituting the values:
\[ A = -\frac{(13.89)^2}{2 \times 0.4} = -\frac{193.6121}{0.8} \approx -242.01 \, \text{m/s}^2 \]

### Step 3: Calculate the stopping distance at \( U_2 \)
Now we will find the stopping distance \( S_2 \) when the automobile is traveling at \( U_2 = 25 \, \text{m/s} \). Again using the equation of motion:
\[ 0 = U_2^2 + 2AS_2 \implies S_2 = -\frac{U_2^2}{2A} \]

Substituting the values:
\[ S_2 = -\frac{(25)^2}{2 \times -242.01} = \frac{625}{484.02} \approx 1.29 \, \text{m} \]

### Step 4: Convert \( S_2 \) to centimeters
Since we need the answer in centimeters:
\[ S_2 = 1.29 \, \text{m} \times 100 = 129 \, \text{cm} \]

### Final Answer
The minimum stopping distance when the automobile is traveling at \( 90 \, \text{km/h} \) is approximately **129 cm**.

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