A `2-m` wide truck is moving with a uniform speed `v_(0)=8 ms^(-1)` along a straight horizontal road. `A` pedestrian starts to cross the road with a uniform speed `v` when the truck is `4 m` away from him, The minimum value of `v` so that he can cross the road safely is .

Options

A. `2.62m//s`
B. `4.6m//s`
C. `3.57m//s`
D. `1.414m//s`

Answer

To solve the problem, we need to determine the minimum speed \( v \) of the pedestrian so that he can cross the road safely before the truck reaches him. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the scenario
- The truck is 2 meters wide and is moving towards the pedestrian at a speed of \( v_0 = 8 \, \text{m/s} \).
- The pedestrian starts crossing the road when the truck is 4 meters away from him.

### Step 2: Calculate the time taken by the truck to reach the pedestrian
- The distance from the truck to the pedestrian is 4 meters.
- The speed of the truck is \( v_0 = 8 \, \text{m/s} \).
- The time \( t_t \) taken by the truck to reach the pedestrian can be calculated using the formula:
\[ t_t = \frac{\text{Distance}}{\text{Speed}} = \frac{4 \, \text{m}}{8 \, \text{m/s}} = 0.5 \, \text{s} \]

### Step 3: Calculate the time required for the pedestrian to cross the road
- The pedestrian needs to cross a distance of 2 meters (the width of the truck).
- Let \( v \) be the speed of the pedestrian.
- The time \( t_p \) taken by the pedestrian to cross the road can be expressed as:
\[ t_p = \frac{\text{Distance}}{\text{Speed}} = \frac{2 \, \text{m}}{v} \]

### Step 4: Set the times equal for safe crossing
- For the pedestrian to cross safely, the time taken by the truck to reach him must be equal to or greater than the time taken by the pedestrian to cross:
\[ t_t \geq t_p \]
Substituting the expressions we found:
\[ 0.5 \, \text{s} \geq \frac{2 \, \text{m}}{v} \]

### Step 5: Solve for the pedestrian's speed \( v \)
- Rearranging the inequality gives us:
\[ v \geq \frac{2 \, \text{m}}{0.5 \, \text{s}} = 4 \, \text{m/s} \]

### Conclusion
- The minimum speed \( v \) of the pedestrian so that he can cross the road safely is:
\[ v = 4 \, \text{m/s} \]