Question

1. If a body covers three consecutive equal distances with speeds v1. v2 and
v3 then average speed of the body is:
*) (v1 + V2 -13)/3
b) 2v1v2/(vl+V2) -1/v3
c) 3v1v2v3/(vl-v2-v3)
d) 3v1v2v3/(vlv2+ vlv3 -v2v3)
body is travelling at constant sneed in a circle it has:
1​

Answer 1

$\textsf{Given that the body cover}\\ \textsf{equal distances with speed} \\ \textsf{ v_{1}, v_{2} and v_{3} } \\ \\ \textsf{Let the distance travelled}\\ \textsf{be d, We have} \\ \\ \\ \textsf{\large{\underline{Case 1:}}} \\ \hookrightarrow \sf Distance \: travelled = d\\ \hookrightarrow \sf Speed = v_{1} \\ \\ \sf We know, \\ \quad \boxed{\sf Time = \dfrac{Distance}{Speed}} \\ \\ \\ \sf \Rightarrow \quad t_{1} = \dfrac{d}{v_{1}} \qquad \hdots (1) \\ \\ \\ \textsf{\large{\underline{Case 2:}}}$$\hookrightarrow \sf Distance \: travelled = d \\ \hookrightarrow Speed = }\: v_{2} \\ \\ \sf We have, \\ \\ \Rightarrow \quad \sf Time \: taken = \dfrac{Distance}{Speed} \\ \\ \\ \Rightarrow \quad \sf t_{2} = \dfrac{d}{v_{2}} \quad \hdots (2) \\ \\ \\ \textsf{\large{\underline{Case 3:}}} \\ \hookrightarrow \sf Distance \: travelled = d \\ \hookrightarrow \sf Speed = v_{3} \\ \\ \sf We know, \\ \\ \Rightarrow \quad \sf Time \: taken = \dfrac{Distance}{Speed}$$\Rightarrow \quad \sf t_{3} = \dfrac{d}{v_{3}} \quad \hdots (3) \\ \\ \\ \sf Now, \\ \\ \quad \boxed{\sf Average \: Speed = \dfrac{Total \: distance \: travelled}{Total \: time \: taken}} \\ \\ \\ \Rightarrow \quad \sf Speed_{avg} = \dfrac{d + d + d}{t_{1} + t_{2} + t_{3}} \\ \\ \\ \Rightarrow \quad \sf Speed_{avg} = \dfrac{3d}{\dfrac{d}{v_{1}} + \dfrac{d}{v_{2}} + \dfrac{d}{v_{3}}} \quad \bigg[ from (1), (2) \& (3) \bigg]$$\Rightarrow \quad \sf Speed_{avg} = \dfrac{3d}{\dfrac{d(v_{2}v_{3})+ d(v_{1}v_{3}) + d(v_{1}v_{2})}{v_{1}v_{2}v_{3}}} \\ \\ \\ \Rightarrow \quad \sf Speed_{avg} = \dfrac{3\not{d }\cdot v_{1}v_{2}v_{3}}{\not{d}(v_{2}v_{3} + v_{1}v_{3} + v_{1}v_{2})} \\ \\ \\ \Rightarrow \quad \sf Speed_{avg} = \dfrac{3v_{1}v_{2}v_{3}}{v_{1}v_{2} + v_{2}v_{3} + v_{1}v_{3}}$$\textsf{Therefore, The average speed} \\ \textsf{of body is \dfrac{3v_{1}v_{2}v_{3}}{v_{1}v_{2} + v_{2}v_{3} + v_{3}v_{1}} }$