Question

A person travels x distance with velocity v₁ and then x distance with velocity v₂ in the same direction. The average velocity of the person is v, then the relation between v, v₁ and v₂ will be

• A. \(v=\frac{v_1+v_2}{2}\)
• B. V=V₁+V₂
• C. \(\frac{1}{v}=\frac{1}{v_1}+\frac{1}{v_2}\)<
• D. \(\frac{2}{v}=\frac{1}{v_1}+\frac{1}{v_2}\)

Answer 1

The Correct Option is D

Let the person travels the first \( x \) distance with velocity \( v_1 \), and the next \( x \) distance with velocity \( v_2 \). The time taken for the first part of the journey is: \[ t_1 = \frac{x}{v_1} \] The time taken for the second part of the journey is: \[ t_2 = \frac{x}{v_2} \] The total displacement is \( x + x = 2x \) and the total time is: \[ t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2} = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \] The average velocity is given by: \[ v = \frac{\text{Total displacement}}{\text{Total time}} = \frac{2x}{t_1 + t_2} = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} = \frac{2}{\left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \] Thus, the relation is: \[ \frac{2}{v} = \frac{1}{v_1} + \frac{1}{v_2} \]