Question

A train travels with a speed of $60km{{h}^{-1}}$ from station A to station B and then comes back with a speed of $80km{{h}^{-1}}$ from station B to station A. i) The average speed and ii) the average velocity of the train

A.)$70km{{h}^{-1}}$
B.)$\dfrac{480}{7}km{{h}^{-1}}$
C.)$70m{{s}^{-1}}$
D.)$\dfrac{480}{7}m{{s}^{-1}}$

Answer 1

Hint: Define scalar and vector quantity. Define speed and velocity. obtain the mathematical expression for speed and velocity. find the displacement and the distance covered by the train. Put these values on the obtained mathematical expression to get the required solution.

Complete step by step answer:

Let the distance between station A and station B is x.
Now, speed can be defined as the distance covered per unit time and velocity can be defined as the displacement of the object per unit time.
The difference in speed and velocity is that speed is a scalar quantity and velocity is a vector quantity.
So, we can say that the distance is a scalar quantity and displacement is a vector quantity.
Now, the train travels to station B from station A and comes back to station A from station B.
So, the total distance covered by the train will be $x+x=2x$
Now, the time taken by the train when moving from station A to station B with speed $60km{{h}^{-1}}$ is,

${{t}_{1}}=\dfrac{x}{60}h$

Again, the time taken by the train when moving from station B to station A with speed $80km{{h}^{-1}}$ is,

${{t}_{2}}=\dfrac{x}{80}h$

Total time taken by the train is,

$\begin{aligned} & t={{t}_{1}}+{{t}_{2}} \\\ & t=\dfrac{x}{60}+\dfrac{x}{80} \\\ & t=\dfrac{7x}{240}h \\\ \end{aligned}$

So, the average speed of the train is,
Average speed

$\begin{aligned} & =\dfrac{\text{distance}}{\text{time}} \\\ & =\dfrac{2x}{\dfrac{7x}{240}} \\\ & =\dfrac{480}{7}km{{h}^{-1}} \\\ \end{aligned}$

So, the average speed of the train will be, $\dfrac{480}{7}km{{h}^{-1}}$
Again, the train comes back to the starting point i.e. the train moves from station A and again comes back to station A. so, the displacement of the train will be 0.

So,
Average velocity

$\begin{aligned} & =\dfrac{\text{displacement}}{\text{time}} \\\ & =\dfrac{0}{\dfrac{7x}{240}} \\\ & =0 \\\ \end{aligned}$

So, the average velocity of the train is zero.
The correct option is (B).

Note:
The scalar quantities have only magnitude, not direction. On the other hand, the vector quantities have both the magnitude and the direction. In the above question speed is a scalar quantity which only has magnitude. That’s why we get a value for speed. But velocity has both direction and magnitude. That’s why we get zero velocity for the train.