An object moves with speed $v_{1}, v_{2}$, and $v_{3}$ along a line segment $\mathrm{AB}, \mathrm{BC}$ and CD respectively as shown in figure. Where $\mathrm{AB}=\mathrm{BC}$ and $\mathrm{AD}=3 \mathrm{AB}$, then average speed of the object will be :

(C) $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2}+v_{2} v_{3}+v_{3} v_{1}}$

(A) $\frac{\left(v_{1}+v_{2}+v_{3}\right)}{3}$

(B) $\frac{v_{1} v_{2} v_{3}}{3\left(v_{1} v_{2}+v_{2} v_{3}+v_{3} v_{1}\right)}$

(D) $\frac{\left(v_{1}+v_{2}+v_{3}\right)}{3 v_{1} v_{2} v_{3}}$

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Kinematics

An object moves with speed

v_{1}, v_{2}

, and

v_{3}

along a line segment

AB, BC

and CD respectively as shown in figure. Where

AB = BC

and

AD = 3 AB

, then average speed of the object will be :

(A) $\frac{( v _{1} + v _{2} + v _{3} )}{3}$
(B) $\frac{v _{1} v _{2} v _{3}}{3 ( v _{1} v _{2} + v _{2} v _{3} + v _{3} v _{1} )}$
(C) $\frac{3 v _{1} v _{2} v _{3}}{v _{1} v _{2} + v _{2} v _{3} + v _{3} v _{1}}$ ✓ Correct
(D) $\frac{( v _{1} + v _{2} + v _{3} )}{3 v _{1} v _{2} v _{3}}$

Answer: (C)

With

A B = B C = x

and

A D = 3 x

, the last stretch is

C D = A D - A B - B C = x

, so the three segments are of equal length

x

. The time spent over them is

\frac{x}{v _{1}} + \frac{x}{v _{2}} + \frac{x}{v _{3}}

, while the whole path covered is

3 x

. The average speed is the total distance divided by the total time,

⟨ v ⟩ = \frac{3 x}{\frac{x}{v _{1}} + \frac{x}{v _{2}} + \frac{x}{v _{3}}} = \frac{3 v _{1} v _{2} v _{3}}{v _{1} v _{2} + v _{2} v _{3} + v _{3} v _{1}} .

JEE Main · 2023 · 01 Feb 2023 · Shift 1 · Single correct option