Question

A train travels with a speed of \[60~km/~h\] from station A to station B and then comes back with a speed \[80~km/~h\] from station B to station A. Find,

 i)The average speed and
 ii)The average velocity of the train.

Answer 1

Hint: Average speed of an object is given as the ratio of total distance to total time. Here assume distance between the stations as \[\text{x}\]. Then find the time taken to travel from A to B and B to A with the given values of speed. Substituting values of time in the speed equation will give us the average speed of the train.
Formula used:
\[\text{Time = }\dfrac{\text{distance}}{\text{speed}}\]
\[\text{Average speed=}\dfrac{\text{Total distance}}{\text{Total time}}\]
\[\text{Average velocity=}\dfrac{\text{Total displacement}}{\text{Total time}}\]

Complete answer:
Given,
When travelling from A to B,
 \[\text{Speed of the train = 60 km/h}\]
\[\text{Time = }\dfrac{\text{distance}}{\text{speed}}\]
Assume that the distance between stations A to B is\[x~Km\].
 Then,
\[\text{ }\!\!~\!\!\text{Time taken to travel from A to B =}\dfrac{\text{x}}{\text{60}}\text{hr}\]---------1
When travelling from B to A,
\[\text{Speed of the train = 80 km/h}\]
\[\text{ }\!\!~\!\!\text{ Time taken to travel from B to A =}\dfrac{\text{x}}{\text{80}}\text{hr}\]--------2
\[\text{Total distance covered by train = x+x = 2x}\] --------- 3
We know that,
\[\text{Average speed=}\dfrac{\text{Total distance}}{\text{Total time}}\]
Substituting 1, 2 and 3 in above equation, we get,
\[\text{Average speed}=\text{ }\dfrac{x+x}{\dfrac{x}{60}+\dfrac{x}{80}}=\dfrac{2x}{\dfrac{4x+3x}{240}}=68.57\text{ }km/hr\]
 We have,
\[\text{Average velocity=}\dfrac{\text{Total displacement}}{\text{Total time}}\]
 Here the total displacement is zero. Hence the average velocity is zero

So, the correct answer is “Option A”.

Note:
Alternate method to find the average speed:
Let the speed of the train are x & y. Then, we have,
 \[\text{Average speed}=\dfrac{2xy}{x+y}\]
Here,
\[\text{Speed of the train, x = 60 km/hr}\]
\[\text{Speed of the train, y = 80 km/hr}\]
Then,
\[\text{Average speed}=\dfrac{2\times 60\times 80}{60+80}=\dfrac{9600}{140}=68.57Km/hr\]