Question

A particle travels half of the distance of a straight journey with speed $6{m}/{s}$. The remaining part of the journey is covered with speed $2{m}/{s}$ for half of the time of the remaining journey and with speed $4{m}/{s}$ for the other half of the time. The average speed of the particle is:
A.$3{m}/{s}$
B.$4{m}/{s}$
C.$\dfrac {3}{4}{m}/{s}$
D.$5{m}/{s}$

Answer 1

Hint: 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 and third part 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:
$Speed= \dfrac {Distance}{time}$

Complete answer:
Let the total distance be d.
In the first part of the journey, particles travel half of the distance with $6{m}/{s}$ speed.
Distance-time relation is given by,
$Speed= \dfrac {Distance}{time}$
Rearranging the above expression we get,
 $time= \dfrac {Distance}{time}$
Now, substituting the values we get,
${t}_{1} =\dfrac {\dfrac{d}{2}}{6}$
$\Rightarrow {t}_{1} = \dfrac {d}{12}sec$
The second part of the journey is again divided into two parts of equal time.
$\Rightarrow {t}_{2}={t}_{3}$
Let the distance covered in the second part of the journey be y.
$\Rightarrow \dfrac {x}{2}= \dfrac {\dfrac{d}{2} - x}{4}$
$\Rightarrow 4x= 2(\dfrac{d}{2} – x)$
$\Rightarrow x= \dfrac{d}{6}m$
$\Rightarrow {t}_{2}={t}_{3}= \dfrac{ \dfrac{d}{6}}{2}$
$\Rightarrow {t}_{2}={t}_{3}= \dfrac {d}{12}sec$
Total time taken for the journey is given by,
$T= {t}_{1}+{t}_{2}+{t}_{3}$
Now, substituting the values we get,
$ T= \dfrac {d}{12}+ \dfrac {d}{12}+ \dfrac {d}{12}$
$\Rightarrow T= 3\dfrac {d}{12}$
$\Rightarrow T=\dfrac {d}{4}$
Average speed of the particle is,
$Speed= \dfrac {Distance}{time}$
Substituting the values we get,
$Speed= \dfrac {d}{T}$
$\Rightarrow Speed= \dfrac {d}{\dfrac {d}{4}}$
$\Rightarrow Speed = 4{m}/{s}$
Thus, the average speed of the particle is $4{m}/{s}$.

 Hence, the correct answer is option B i.e. $4{m}/{s}$.

 Note:
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.