A) \[\frac{2{{v}_{d}}{{v}_{u}}}{{{v}_{d}}+{{v}_{u}}}\]
B) \[\sqrt{{{v}_{u}}{{v}_{d}}}\]
C) \[\frac{{{v}_{d}}{{v}_{u}}}{{{v}_{d}}+{{v}_{u}}}\]
D) \[\frac{{{v}_{u}}+{{v}_{d}}}{2}\]
Correct Answer: A
Solution :
We define \[\text{Average}\,\text{speed=}\frac{\text{Distance}\,\text{travelled}}{\text{Time}\,\text{taken}}\text{=}\frac{\text{d}}{T}\] Let\[{{t}_{1}}\]and \[{{t}_{2}}\]be times taken by the car to go from X to Y and then from Y to X respectively. Then\[{{t}_{1}}+{{t}_{2}}=\left[ \frac{XY}{{{v}_{u}}} \right]+\left[ \frac{XY}{{{v}_{d}}} \right]=XY\left( \frac{{{v}_{u}}+{{v}_{d}}}{{{v}_{u}}{{v}_{d}}} \right)\] Total distance travelled \[d=XY+XY=2XY\] Therefore, average speed of the car for this round trip is \[{{v}_{av}}=\frac{2XY}{XY\left( \frac{{{v}_{u}}+{{v}_{d}}}{{{v}_{u}}+{{v}_{d}}} \right)}\]or \[{{v}_{av}}=\frac{2{{v}_{u}}{{v}_{d}}}{{{v}_{u}}+{{v}_{d}}}\]
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