Question

A body describes the first half of the total distance with velocity v₁ and the second half with velocity v₂. The average velocity is :

(A) \(\cfrac{v_1 +v_2}{2}\)

(B) \(\cfrac{1}{v_1}+\cfrac{1}{v_2}\)

(C) \(\cfrac{v_1 \,v_2}{v_1+v_2}\)

(D) \(\cfrac{2v_1\ v_2}{v_1+v_2}\)

Answer 1

Correct option is (D) \(\frac{2v_1v_2}{v_1 + v_2}\)

For first half of the total distance:

Let the distance covered be x, in time t₁ with velocity v₁.

From the formula,

\(v_1 = \frac x{t_1}\)

\(t_1 = \frac x{v_1}\)

For second half of the total distance:

Let the distance covered be x, in time t₂ with velocity v₂.

From the formula,

\(v_2 = \frac x{t_2}\)

\(t_2 = \frac x{v_2}\)

Therefore, 

Total time taken = t₁ + t₂

We know that,

Average velocity = \(\mathrm{\frac{Total \ displacement}{total\ time} v_{avg}}\)

\(=\mathrm{\frac{2x}{t_1 + t_2} v_{avg}}\)

\(=\mathrm{\frac{2x}{\frac x{v_1} + \frac x{v_2}} v_{avg}}\)

\(\mathrm{= \frac{2x}{x\left(\frac{v_1 + v_2}{v_1v_2}\right)} v_{avg}}\)

\(\mathrm{= \frac{2v_1v_2}{v_1 + v_2}}\)

Answer 2

 Correct option is (D) \(\cfrac{2v_1\ v_2}{v_1+v_2}\)

Answer 3

let's distance travelled by a body with V1 is X by 2 and distance travelled by of the distance by the body x by 2 with velocity V2. so T1 is equal to X by 2 V1 and t2 is equal to X by 2 V2. average velocity is equal to X by X by 2 V1 + X / 2 V2 then take LCM and solve it.