An object moves with speeds , and along the line segments AB, BC and CD respectively as shown in the figure. Where AB = BC and AD = 3AB, then the average speed of the object will be: [2023]

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Q.
An object moves with speeds $v_{1}$ , $v_{2}$ and $v_{3}$ along the line segments AB, BC and CD respectively as shown in the figure. Where AB = BC and AD = 3AB, then the average speed of the object will be: [2023]

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

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

3 $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$

4 $\frac{(v_{1} + v_{2} + v_{3})}{3 v_{1} v_{2} v_{3}}$

Ans.
(3)

$A B = x and B C = x$

$2 x + C D = 3 x$

$\Rightarrow C D = x$

$< v > = \frac{3 x}{\frac{x}{v_{1}} + \frac{x}{v_{2}} + \frac{x}{v_{3}}} = \frac{3 v_{1} v_{2} v_{3}}{v_{2} v_{3} + v_{1} v_{3} + v_{1} v_{2}}$

Solution

Q.
An object moves with speeds $v_{1}$ , $v_{2}$ and $v_{3}$ along the line segments AB, BC and CD respectively as shown in the figure. Where AB = BC and AD = 3AB, then the average speed of the object will be: [2023]

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

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

3 $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$

4 $\frac{(v_{1} + v_{2} + v_{3})}{3 v_{1} v_{2} v_{3}}$

Ans.
(3)

$A B = x and B C = x$

$2 x + C D = 3 x$

$\Rightarrow C D = x$

$< v > = \frac{3 x}{\frac{x}{v_{1}} + \frac{x}{v_{2}} + \frac{x}{v_{3}}} = \frac{3 v_{1} v_{2} v_{3}}{v_{2} v_{3} + v_{1} v_{3} + v_{1} v_{2}}$

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Solution

Q. An object moves with speeds v1, v2 and v3 along the line segments AB, BC and CD respectively as shown in the figure. Where AB = BC and AD = 3AB, then the average speed of the object will be: [2023]

1 (v1+v2+v3)3

2 v1v2v33(v1v2+v2v3+v3v1)

3 3v1v2v3v1v2+v2v3+v3v1

4 (v1+v2+v3)3v1v2v3

Ans. (3) AB=x and BC=x 2x+CD=3x ⇒ CD=x <v>=3xxv1+xv2+xv3=3v1v2v3v2v3+v1v3+v1v2

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Q.
An object moves with speeds $v_{1}$ , $v_{2}$ and $v_{3}$ along the line segments AB, BC and CD respectively as shown in the figure. Where AB = BC and AD = 3AB, then the average speed of the object will be: [2023]

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

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

3 $\frac{3 v_{1} v_{2} v_{3}}{v_{1} v_{2} + v_{2} v_{3} + v_{3} v_{1}}$

4 $\frac{(v_{1} + v_{2} + v_{3})}{3 v_{1} v_{2} v_{3}}$

Ans.
(3)

$A B = x and B C = x$

$2 x + C D = 3 x$

$\Rightarrow C D = x$

$< v > = \frac{3 x}{\frac{x}{v_{1}} + \frac{x}{v_{2}} + \frac{x}{v_{3}}} = \frac{3 v_{1} v_{2} v_{3}}{v_{2} v_{3} + v_{1} v_{3} + v_{1} v_{2}}$