A train moving at a constnt velocity of 54 km/hr moves eastwards for 30 minuts, then due north with the same speed for 40 minutes. What is the average velocity of the train during this run? (in km/hr)

Options

A. 30
B. 35
C. `38.6`
D. `49.3`

Answer

To solve the problem, we need to calculate the average velocity of the train after it moves eastwards and then northwards. Let's break it down step by step:

### Step 1: Convert the speed from km/hr to a usable unit
The train's speed is given as 54 km/hr. Since the time intervals are given in minutes, we will convert the time into hours for consistency.

### Step 2: Calculate the displacement in the eastward direction
The train moves eastwards for 30 minutes. We convert 30 minutes into hours:
\[ \text{Time in hours} = \frac{30 \text{ minutes}}{60} = 0.5 \text{ hours} \]
Now, we calculate the displacement in the eastward direction:
\[ \text{Displacement}_{\text{east}} = \text{Speed} \times \text{Time} = 54 \text{ km/hr} \times 0.5 \text{ hr} = 27 \text{ km} \]

### Step 3: Calculate the displacement in the northward direction
Next, the train moves northwards for 40 minutes. We convert 40 minutes into hours:
\[ \text{Time in hours} = \frac{40 \text{ minutes}}{60} = \frac{2}{3} \text{ hours} \]
Now, we calculate the displacement in the northward direction:
\[ \text{Displacement}_{\text{north}} = \text{Speed} \times \text{Time} = 54 \text{ km/hr} \times \frac{2}{3} \text{ hr} = 36 \text{ km} \]

### Step 4: Calculate the resultant displacement
The total displacement can be found using the Pythagorean theorem since the two displacements are perpendicular to each other:
\[ \text{Resultant Displacement} = \sqrt{(\text{Displacement}_{\text{east}})^2 + (\text{Displacement}_{\text{north}})^2} \]
Substituting the values:
\[ \text{Resultant Displacement} = \sqrt{(27 \text{ km})^2 + (36 \text{ km})^2} = \sqrt{729 + 1296} = \sqrt{2025} = 45 \text{ km} \]

### Step 5: Calculate the total time taken
The total time taken for the journey is:
\[ \text{Total Time} = 0.5 \text{ hr} + \frac{2}{3} \text{ hr} = \frac{3}{6} + \frac{4}{6} = \frac{7}{6} \text{ hr} \]

### Step 6: Calculate the average velocity
Average velocity is given by the formula:
\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \]
Substituting the values:
\[ \text{Average Velocity} = \frac{45 \text{ km}}{\frac{7}{6} \text{ hr}} = 45 \text{ km} \times \frac{6}{7} = \frac{270}{7} \approx 38.57 \text{ km/hr} \]

### Final Answer
The average velocity of the train during this run is approximately **38.57 km/hr**.

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