Overtaking Speed Problem Setup

This problem requires calculating the speed of a truck based on a bus overtaking it. Key concepts involve relative speed and unit conversions between km/h and m/s.

Given Motion Data

• Bus Speed ($v_b$) = 45 km/h
• Initial Distance = 150 m
• Overtake Time ($t$) = 30 s
• Direction: Same

Speed Calculation Steps

Step 1: Convert Bus Speed to m/s

Convert the bus speed from kilometers per hour (km/h) to meters per second (m/s) for calculation consistency.

The conversion formula is: $v (\text{m/s}) = v (\text{km/h}) \times \frac{5}{18}$

$ v_b = 45 \times \frac{5}{18} = \frac{225}{18} = 12.5 \text{ m/s} $

Step 2: Determine Relative Speed

When objects move in the same direction, their relative speed ($v_{rel}$) is the difference between their speeds ($v_b - v_t$). The distance the bus needs to cover relative to the truck is the initial separation distance.

Using the formula: Distance = Relative Speed × Time

$ 150 \text{ m} = v_{rel} \times 30 \text{ s} $

Solve for relative speed:

$ v_{rel} = \frac{150 \text{ m}}{30 \text{ s}} = 5 \text{ m/s} $

Step 3: Calculate Truck Speed in m/s

Use the relative speed definition for objects moving in the same direction: $v_{rel} = v_b - v_t$. Rearrange to find the truck's speed ($v_t$).

$ v_t = v_b - v_{rel} $

Substitute the known values:

$ v_t = 12.5 \text{ m/s} - 5 \text{ m/s} = 7.5 \text{ m/s} $

Step 4: Convert Truck Speed to km/h

Convert the calculated truck speed from meters per second (m/s) back to kilometers per hour (km/h) to match the option format.

The conversion formula is: $v (\text{km/h}) = v (\text{m/s}) \times \frac{18}{5}$

$ v_t = 7.5 \times \frac{18}{5} = \frac{7.5}{5} \times 18 = 1.5 \times 18 = 27 \text{ km/h} $

Truck Speed Result

The speed of the truck is 27 km/h.