Question

A particle travels half the distance with a velocity Vo.The remaining half was travelled with V1 for half the time and with V2 for the other half of time . Find the average velocity of the particle for complete motion.

Answer 1

★ Given :

A particle travels half the distance with a velocity Vo.The remaining half was travelled with V1 for half the time and with V2 for the other half of time . Find the average velocity of the particle for complete motion.

★ Solution :

Figure:

$\setlength{\unitlength}{30} \begin{picture}(6,6) \put(1,1){\line(1,0){9}}\put(1,1){\circle*{0.15}}\put(4,1){\circle*{0.15}}\put(7,1){\circle*{0.15}}\put(10,1){\circle*{0.15}}\put(2.25,1.25){\vector( - 1,0){1.25}}\put(2.25,1.25){\vector(1,0){1.7}}\put(0.8,0.5){ \bf A }\put(3.9,0.5){ \bf B }\put(9.8,0.5){ \bf C }\put(2,1.5){ \bf V_0 }\put(5.5,1.25){ \bf V_1 }\put(8.5,1.25){ \bf V_2 }\put(5.5,0.75){ \bf t }\put(8.5,0.75){ \bf t }\end{picture}$

AB = BC as half the distance is covered in both parts..

We know that ,

The formula for average speed when a body is travelling half the distance with some velocity and other half with some velocity =

$V _ {avg} = \dfrac{2V_1 V_2}{V_1 + V_2}$

In this case ,

$V_1 = v_0$

$V_2 = average \:of \: v_1 \: and\: v_2$

We know that,

Average speed when the body travels with two different velocities for equal time intervals =

$\dfrac{v_1 + v_2}{2}$

$by \:putting \:the \:value \:of\: V_2\: as \: \dfrac{v_1 + v_2}{2}$

$= \dfrac{ 2v_0 \times \dfrac{v_1 + v_2}{2}}{v_0 + \dfrac{v_1 + v_2}{2}}$

$= \dfrac{v_0 ( v_1 + v_2) }{\dfrac{2v_0 + v_1 + v_2 }{2}}$

$= \dfrac{2v_0(v_1 + v_2)}{2v_0 + v_1 + v_2}$

$\purple{ V_{avg} = \dfrac{2v_0(v_1 + v_2)}{2v_0 + v_1 + v_2}}$