Answer :
Answer:
To find the average speed of the train for the entire journey, we need to calculate both the average velocity and the average speed. Here's how we can proceed:
Given:
- Speed of the train from A to B = 40 km/h
- Speed of the train from B to A (return journey) = 60 km/h
### a. Average Velocity of the Train:
Average velocity \( \bar{v} \) is given by the total displacement divided by the total time taken.
1. **Calculate the total displacement:**
Since the train travels from A to B and then returns from B to A, the total displacement is zero because it returns to its starting point.
\[ \text{Total displacement} = 0 \]
2. **Calculate the total time taken:**
Let's assume the distance between A and B is \( d \) kilometers.
- Time taken from A to B: \( t_1 = \frac{d}{40} \) hours
- Time taken from B to A: \( t_2 = \frac{d}{60} \) hours
Total time for the round trip:
\[ T_{\text{total}} = t_1 + t_2 = \frac{d}{40} + \frac{d}{60} \]
\[ T_{\text{total}} = \frac{3d}{120} + \frac{2d}{120} = \frac{5d}{120} \]
\[ T_{\text{total}} = \frac{d}{24} \]
3. **Calculate the average velocity:**
Average velocity \( \bar{v} \) is:
\[ \bar{v} = \frac{\text{Total displacement}}{\text{Total time}} = \frac{0}{\frac{d}{24}} = 0 \]
Therefore, the average velocity of the train for the entire journey is \( \boxed{0} \).
### b. Average Speed of the Train:
Average speed \( \bar{v}_{\text{avg}} \) is given by the total distance traveled divided by the total time taken.
1. **Calculate the total distance traveled:**
Total distance \( D_{\text{total}} \) for the round trip:
\[ D_{\text{total}} = d + d = 2d \]
2. **Calculate the total time taken (already calculated):**
Total time for the round trip:
\[ T_{\text{total}} = \frac{d}{24} \]
3. **Calculate the average speed:**
Average speed \( \bar{v}_{\text{avg}} \) is:
\[ \bar{v}_{\text{avg}} = \frac{D_{\text{total}}}{T_{\text{total}}} = \frac{2d}{\frac{d}{24}} = 2d \cdot \frac{24}{d} \]
\[ \bar{v}_{\text{avg}} = 48 \]
Therefore, the average speed of the train for the entire journey is \( \boxed{48} \) km/h.
Answer:
Net displacement of the train in coming back to the first station is zero i.e.
S
=
0
.
Average velocity
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a
v
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Average speed of the train when in moving equal distances is given by
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V
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a
v
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=
2
v
1
v
2
v
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v
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=
2
×
40
×
60
40
+
60
=
48
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