Question

A particle covers half of the distance with speed 'u' and other half with speed 'v'. Find average velocity.​

Answer 1

$\red{\bigstar}\underline{\underline{\textsf{\textbf{ Given :- }}}}$

• A particle covers half of the distance with speed 'u' and other half with speed 'v'.

$\red{\bigstar}\underline{\underline{\textsf{\textbf{ To Find :- }}}}$

• The average velocity of the particle .

$\red{\bigstar}\underline{\underline{\textsf{\textbf{ Solution :- }}}}$

We need to find the average velocity of the journey .We know that average velocity is defined as the total displacement travelled by total time taken. Therefore let us take ,

• The person took a time of t₂ to travel second half and a time of t₁ to travel first half of the motion.

For the figure refer to the attachment . Now we know that , velocity is equal to displacement by time. So that ,

$\tt\dashrightarrow t_1 = \dfrac{ x}{u}\\\\\\\tt\dashrightarrow t_2 = \dfrac{x}{v}$

❒ So that ,

$\tt\dashrightarrow Avg . \ Velocity =\dfrac{ Total \ displacement\ travelled }{Total \ time\ taken } \\\\\\\tt\dashrightarrow v_{(avg)}= \dfrac{ x+x}{t_1+t_2} \\\\\\\tt\dashrightarrow v_{(avg)}= \dfrac{ 2x}{\dfrac{x}{u}+\dfrac{x}{v}} \\\\\\\tt\dashrightarrow v_{(avg)}= \dfrac{2x}{\dfrac{xu+xv}{uv}} \\\\\\\tt\dashrightarrow v_{(avg)}= \dfrac{(2x) * ( uv)}{x( u+v)} \\\\\\\tt\dashrightarrow \underset{\blue{\sf Required\ Answer}}{\underbrace{\boxed{\pink{\frak{Velocity_{(avg)}= \dfrac{ 2uv}{u+v}}}}}}$

$\rule{200}2$