Question

An object moves with speed $v_1, v_2$ and $v_3$ along a line segment $AB , BC$ and $C D$ respectively as shown in figure Where $AB = BC$ and $AD =3 AB$, then average speed of the object will be:

• A. $\frac{\left(v_1+v_2+v_3\right)}{3}$
• B. $\frac{3 v_1 v_2 v_3}{\left(v_1 v_2+v_2 v_3+v_3 v_1\right)}$
• C. $\frac{\left(v_1+v_2+v_3\right)}{3 v_1 v_2 v_3}$
• D. $\frac{v_1 v_2 v_3}{3\left(v_1 v_2+v_2 v_3+v_3 v_1\right)}$

Answer 1

The Correct Option is B

To find the average speed of the object moving along the path \(AB \to BC \to CD\), we start by analyzing the given information:

• The lengths \(AB = BC = x\).
• The total length \(AD = 3x\).
• Velocities along the segments are \(v_1, v_2, v_3\) respectively.

The formula for average speed is given by:

\[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]

Calculating Total Distance:

• Total Distance \(= AD = 3x\)

Calculating Total Time:

• Time to traverse \(AB\): \(t_1 = \frac{x}{v_1}\)
• Time to traverse \(BC\): \(t_2 = \frac{x}{v_2}\)
• Time to traverse \(CD\): \(t_3 = \frac{(AD - AB - BC)}{v_3} = \frac{x}{v_3}\)
• Total Time \(= t_1 + t_2 + t_3 = \frac{x}{v_1} + \frac{x}{v_2} + \frac{x}{v_3}\)

Substituting these values into the average speed formula:

\[ \text{Average Speed} = \frac{3x}{\frac{x}{v_1} + \frac{x}{v_2} + \frac{x}{v_3}} = \frac{3x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} = \frac{3}{\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}} \]

Simplifying the expression further:

\[ = \frac{3}{\frac{v_2 v_3 + v_3 v_1 + v_1 v_2}{v_1 v_2 v_3}} = \frac{3 v_1 v_2 v_3}{v_2 v_3 + v_3 v_1 + v_1 v_2} \]

Thus, the average speed of the object is:

• \(\frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1}\)

The correct answer is: \(\frac{3 v_1 v_2 v_3}{\left(v_1 v_2+v_2 v_3+v_3 v_1\right)}\)