Question

A particle moves along x-axis with speed $6{\text{ m/s}}$ for the first half distance of a journey and the second half distance with the speed ${\text{3 m/s}}$ . The average speed in the total journey is:

Answer 1

In order to answer this question, we need to Use distance time relation to find the time required to cover the first half of the journey. Using the same formula, calculate time required to complete the second half of the journey. Add all the time to get the total time required to complete the journey. Now, from that calculated total time find the average speed.
Formula Used:
${\text{speed = }}\dfrac{{{\text{distance}}}}{{{\text{time}}}}$

Complete answer:
Let the total distance be d.
In the first part of the journey, particles travel half of the distance with $6{\text{ m/s}}$ speed.
Distance-time relation is given by,
${\text{speed = }}\dfrac{{{\text{distance}}}}{{{\text{time}}}}$
Rearranging the above expression we get,
${\text{time = }}\dfrac{{{\text{distance}}}}{{{\text{speed}}}}$
For first half journey,
$d = \dfrac{d}{2}$
$s = 6{\text{ m/s}}$
Now, substituting the values we get,
$\Rightarrow {t_1} = \dfrac{{\dfrac{d}{2}}}{6}$
$\Rightarrow {t_1} = \dfrac{d}{{12}}$
For second half journey,
$d = \dfrac{d}{2}$
$s = 3{\text{ m/s}}$
Now, substituting the values we get,
$\Rightarrow {t_2} = \dfrac{{\dfrac{d}{2}}}{3}$
$\Rightarrow {t_2} = \dfrac{d}{6}$
Total time taken for the journey is given by,
$T = {t_1} + {t_2}$
Now, substituting the values of ${t_1}$ and ${t_2}$ we get,
$\Rightarrow T = \dfrac{d}{{12}} + \dfrac{d}{6}$
$\Rightarrow T = \dfrac{d}{4}$
Average speed of the particle is,
${\text{speed = }}\dfrac{{{\text{distance}}}}{{{\text{time}}}}$
Here,
distance= d
$time = \dfrac{d}{4}$
Now, substituting the values we get,
$\Rightarrow speed = \dfrac{d}{{\dfrac{d}{4}}}$
$\Rightarrow speed = 4{\text{ m/s}}$
Thus, the average speed of the particle is $4{\text{ m/s}}$

Note:
It should be remembered that as the particle covers the distance in an equal interval of time then the average speed of the particle is average of all the speeds. So, using this trick students can cross check their answer by taking the average of all the speeds. If the distance covered is zero then the average speed will be zero. Average speed can never be negative while average velocity can be. The negative sign indicates direction.