Question

Straight distance between a hotel and a railway station is $10km$ , but the circular route is followed by a taxi covering $23km$ in $28{\text{ minutes}}$ . What is the average speed and magnitude of average velocity? Are they equal?

Answer 1

So in this question, we have to find the average speed, and as we know that the average speed is given by total path length by the total time. And for the average velocity, we will use the formula which is given by displacement upon a time.

Formula used:
Average speed is given by,
$Average{\text{ speed = }}\dfrac{{Total{\text{ path length}}}}{{Total{\text{ time}}}}$
Average velocity is given by,
$Average{\text{ velocity = }}\dfrac{{Displacement}}{{{\text{Time}}}}$ .

Complete step by step solution:
So in this question, we have the shortest distance given and we know that the shortest distance is the same as the displacement. Also, we have the time given. Now we will first find the average speed.
Therefore, the formula for the average speed is given by
$Average{\text{ speed = }}\dfrac{{Total{\text{ path length}}}}{{Total{\text{ time}}}}$
Now on substituting the values, we get
$\Rightarrow Average{\text{ speed = }}\dfrac{{28}}{{\left( {28/60} \right)}}km/h$
And on solving the above expression, we will get the equation as
$\Rightarrow Average{\text{ speed = 49}}{\text{.3}}km/h$
Therefore, the average velocity will be equal to $49.3km/hr$
Now we will calculate the average velocity, so the formula will be given by
$Average{\text{ velocity = }}\dfrac{{Displacement}}{{{\text{Time}}}}$
Now on substituting the values, we will get the equation as
$Average{\text{ velocity = }}\dfrac{{10}}{{\left( {28/60} \right)}}km/hr$
And on solving the above expression, we will get the equation as
$\Rightarrow Average{\text{ velocity = 21}}{\text{.4}}km/h$
Therefore, the average velocity will be equal to $21.4km/hr$ .
Now on comparing the values of the average speed and the average velocity we can see that both are different.
So we can say that the two physical quantities given to us are not equal.

Note:
Here the quantity we have is average speed and this quantity is scalar whereas the average velocity is a vector quantity. Also, one important thing which we have to keep in mind is checking the unit. As in this question too we had changed the unit. So always match the unit before solving it.