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 :
Key Idea Average speed of a body in a given time interval is defined as the ratio of distance travelled to the time taken. \[\text{Average}\,\,\text{speed=}\frac{\text{Distance}\,\,\text{traevelled}}{\text{Time}\,\,\text{taken}}\] 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,\[{{t}_{1}}+{{t}_{2}}=\frac{XY}{{{v}_{u}}}+\frac{XY}{{{v}_{d}}}=XY\left( \frac{{{v}_{u}}+{{v}_{d}}}{{{v}_{u}}{{v}_{d}}} \right)\] Total distance travelled \[=XY+XY=2XY\] 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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