The problem asks for the speed of a carriage given information about its visibility to a man walking in the same direction.

Physics Solution: Carriage Speed Calculation

We need to calculate the speed of the carriage using the concepts of relative speed.

Step 1: Convert Units

Convert the given time and distance into consistent units (kilometers and hours).

• Time duration ($t$): 4 minutes = $\frac{4}{60}$ hours = $\frac{1}{15}$ hours.
• Visible distance ($d$): 100 meters = $\frac{100}{1000}$ kilometers = $0.1$ km.
• Man's speed ($v_m$): $3$ km/h.

Step 2: Determine Relative Speed

The man and the carriage are moving in the same direction. Let the carriage's speed be $v_c$. The relative speed ($v_{rel}$) at which the carriage moves away from the man (or the distance between them increases) is the difference between their speeds:

$v_{rel} = v_c - v_m$

During the 4 minutes, the carriage was visible up to a distance of 100 m. This distance represents the relative distance covered during the observation time.

Step 3: Apply Distance-Speed-Time Formula

Use the formula relating distance, relative speed, and time:

$d = v_{rel} \times t$

Substitute the known values:

$0.1 \text{ km} = v_{rel} \times \frac{1}{15} \text{ h}$

Step 4: Calculate Relative Speed

Solve for $v_{rel}$:

$v_{rel} = 0.1 \text{ km} \times 15 \text{ h}^{-1}$

$v_{rel} = 1.5 \text{ km/h}$

Step 5: Calculate Carriage Speed

Now, find the carriage's speed ($v_c$) using the relative speed formula:

$v_c = v_{rel} + v_m$

$v_c = 1.5 \text{ km/h} + 3 \text{ km/h}$

$v_c = 4.5 \text{ km/h}$

The speed of the carriage is $4.5$ km/h.