Question

Body travels the first half of the total distance with velocity v1 and the second half with velocity v2. calculate the average velocity. (1)

Answer 1

Let total displacement be 'x'

Velocity for first half of total displacement= v₁

• Velocity= Displacement/Time
• Time = Displacement/Velocity

Time for first half =  $\sf \dfrac{\frac{x}{2}}{v_1}$ = $\sf \dfrac{x}{2v_1}$

Velocity for second half = v₂

Time for second half = $\sf \dfrac{x}{2v_2}$

Average Velocity: It is defined as total displacement covered to the total time taken

$\to \sf \orange{\tiny{A_{vel}=\dfrac{Total\ displacement}{Total\ time}}\ \; \bigstar}$

$\to \sf A_{Vel}=\dfrac{x}{\frac{x}{2v_1}+\frac{x}{2v_2}}$

$\to \sf A_{Vel}=\dfrac{x}{x\bigg(\frac{1}{2v_1}+\frac{1}{2v_2}\bigg)}$

$\to \sf A_{Vel}=\dfrac{1}{\bigg(\frac{2v_2+2v_1}{4v_1.v_2}\bigg)}$

$\to \sf A_{Vel}=\dfrac{1}{\bigg(\frac{2(v_2+v_1)}{4v_1.v_2}\bigg)}$

$\to \sf A_{Vel}=\dfrac{1}{\bigg(\frac{v_2+v_1}{2v_1.v_2}\bigg)}$

$\to \sf A_{Vel}=\dfrac{2v_1.v_2}{v_1+v_2}\ \; \pink{\bigstar}$

Answer 2

Answer:

$\bull \sf \: Velocity_{avg} = \frac{2v_1v_2}{v_1 + v_2}$

Explanation:

Let total distance will be " 2r ".

→ Half of the distance will be " r ".

• $\sf{ v_1 }$ Velocity for fist half.
• $\sf{ v_2}$ Velocity for second half.

As we know that,

$\boxed{\sf \red{Velocity_{avg} = \frac{Total \: displacement}{Time \: taken} }} \: \: \green\bigstar$

[ Putting values ]

$\leadsto\sf \: Velocity_{avg} = \frac{2r}{ \frac{r}{v_1 } + \frac{r}{v_2} } \\ \\ \leadsto\sf \: Velocity_{avg} = \frac{2r}{ \frac{rv_2 + rv_1}{v1v2} } \\ \\ \leadsto\sf \: Velocity_{avg} = \frac{2r \times \: v_1v_2 }{r(v_1 + v_2)} \\ \\ \leadsto\sf \: Velocity_{avg} = \frac{2 \cancel{r }\times \: v_1v_2 }{ \cancel{r}(v_1 + v_2)} \\ \\ \leadsto \underbrace{\sf \green{ Velocity_{avg} = \frac{2v_1v_2}{v_1 + v_2} } }\: \: \red \bigstar$