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. |
| \[\text{Average}\,\text{speed=}\frac{\text{Distance}\,\text{travelled}}{\text{Time}\,\text{taken}}\] |
| 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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