Question

A train travels from station A to station B with $36km/h$ and returns to A with $54km/h$. What is the average speed?

Answer 1

Hint: Speed is a scalar quantity which has only magnitude and not direction. We know that a speed is the change in the distance with respect to time. Clearly, speed depends on time. Then we can classify speed into average speed and instantaneous speed based on time. Here, we can find the time taken for the individual trips and then sum them to find the speed of the complete trip.

Formula used:
$speed=\dfrac{distance}{time}$

Complete answer:
We know that speed is the change of distance with respect to time, and is given by $speed=\dfrac{distance}{time}$
Average speed is the change in distance $d$ over a period of time$t$, and is given as $u_{avg}=\dfrac{d}{t}$
Whereas, instantaneous speed is the small distance $d\;s$ covered during a small period of time$d\;t$. It is denoted using calculus and is given as $u_{inst}=\dfrac{ds}{dt}$
Let us assume that the train travels a distance $X$ from A to B.
Given that the speed at which the train travels from A to B is $36km/h$. Then the time taken for the same is given as $t_{1}=\dfrac{X}{36}$
Similarly, the speed at which a train travels from B to A is $54km/h$. Then the time taken for the same is given as $t_{2}=\dfrac{X}{54}$
Then, the total time taken for both the trips is given as $t=t_{1}+t_{2}=\dfrac{X}{36}+\dfrac{X}{54}=\dfrac{3X+2X}{108}=\dfrac{5X}{108}$
Then the average speed to cover a distance $2\;X$ during the time $t$ is given as $u=\dfrac{2X}{\dfrac{5X}{108}}=\dfrac{216}{5}=43.2km/hr$
Hence the average speed is $43.2km/h$

Note:
There is a small difference between velocity and speed. Velocity is a vector quantity which has both direction and magnitude, unlike speed which is a scalar quantity and can have only magnitude and not direction. This is one major difference between the both, and it is a physical quantity.