On a foggy day, wo drivers spot in front of each other when 80 metre apart. They were travelling at 70 kmph and 60 kmph. Both apply brakes simultaneously which retard the cars at the rate of `5[m//s^(2)]` which of the following statements is correct ?

Options

A. The collision will be averted
B. The collision will take place
C. They will cross each other
D. They will just collide

Answer

To solve the problem step by step, we need to determine the stopping distances of both cars and check if they collide or not.

### Step 1: Convert Speeds from km/h to m/s
The speeds of the two cars are given as:
- Car 1 (Driver 1): 70 km/h
- Car 2 (Driver 2): 60 km/h

To convert these speeds to meters per second (m/s), we use the conversion factor:
\[ \text{Speed in m/s} = \text{Speed in km/h} \times \frac{5}{18} \]

For Car 1:
\[ u_1 = 70 \times \frac{5}{18} = \frac{350}{18} \approx 19.44 \, \text{m/s} \]

For Car 2:
\[ u_2 = 60 \times \frac{5}{18} = \frac{300}{18} \approx 16.67 \, \text{m/s} \]

### Step 2: Calculate Stopping Distances
Using the third equation of motion:
\[ v^2 = u^2 + 2as \]
Where:
- \( v \) = final velocity (0 m/s, since the cars come to a stop)
- \( u \) = initial velocity
- \( a \) = acceleration (retardation, which is -5 m/s²)
- \( s \) = stopping distance

Rearranging gives us:
\[ s = \frac{v^2 - u^2}{2a} \]

For Car 1:
\[ s_1 = \frac{0 - (19.44)^2}{2 \times (-5)} = \frac{(19.44)^2}{10} \]
Calculating \( (19.44)^2 \):
\[ s_1 = \frac{377.6336}{10} \approx 37.76 \, \text{m} \]

For Car 2:
\[ s_2 = \frac{0 - (16.67)^2}{2 \times (-5)} = \frac{(16.67)^2}{10} \]
Calculating \( (16.67)^2 \):
\[ s_2 = \frac{278.0889}{10} \approx 27.81 \, \text{m} \]

### Step 3: Determine Total Stopping Distance
Now, we need to find the total stopping distance:
\[ s_{total} = s_1 + s_2 \approx 37.76 + 27.81 \approx 65.57 \, \text{m} \]

### Step 4: Compare with Initial Distance
The initial distance between the two cars is 80 m. We need to check if the total stopping distance is less than or equal to this distance:
\[ s_{total} = 65.57 \, \text{m} < 80 \, \text{m} \]

### Conclusion
Since the total stopping distance (65.57 m) is less than the initial distance (80 m), the two cars will not collide.

### Final Answer
**The correct statement is that the two cars will not collide.**

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