Question

A body travels the first half of the total distance with velocity v1 and the second half with a velocity me to calculate the average velocity

Answer 1

Answer:

$v_{av}\ =\ \dfrac{2v_1v_2}{v_1\ +\ v_2}$

Explanation:

Given,

• Speed of the body For the first half= $v_1$
• Speed of the body for the second half  = $v_2$

Let '2s' be the total distance traveled by the body.

In case, the body moving first half of the total distance,

Let $t_1$ be the total time taken by the body.

$\therefore s\ =\ v_1t_1\Rightarrow t_1\ =\ \dfrac{s}{v_1}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,eqn (1)$

In case, the body moving the second half of the total distance,

Let $t_2$ be the total time taken by the body to traveled the second half of the total distance.

$\therefore s\ =\ v_2t_2\\\Rightarrow t_2\ =\ \dfrac{s}{v_2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,eqn (2)$

Therefore from the average velocity formula

$\therefore v_{av}\ =\ \dfrac{s_1\ +\ s_2}{t_1\ +\ t_2}\\\Rightarrow v_{av}\ =\ \dfrac{s\ +\ s}{\dfrac{s}{v_1}\ +\ \dfrac{s}{v_2}}\\\Rightarrow v_{av}\ =\ \dfrac{2s}{\dfrac{sv_2\ +\ sv_1}{v_1v_2}}\\\Rightarow v_{av}\ =\ \dfrac{2sv_1v_2}{s(v_1\ +\ v_2)}\\\Rightarrow v_{av}\ =\ \dfrac{2v_1v_2}{v_1\ +\ v_2}$

Hence the average speed of the body is $\dfrac{2v_1v_2}{v_1\ +\ v_2}$.

Answer 2

Answer:

Average Velocity = 2V1V2/(V1+V2)

Here is the proof:

Let the total distance be x.

So time taken to cover first half =(x/2)/v1=x/(2v1)

Time taken to cover second half=(x/2)/v2=x/(2v2)

Now average velocity=Total Distance/Total time = x/((x/2v1)+(x/2v2)) = 2v1v2/v1+v2