- #1

MatinSAR

- 590

- 182

- Homework Statement
- A car covers half of the road with an average velocity of v, 1/4 of the road with an average velocity of 2v, and 1/8 of the road with an average velocity of 4v, and so on until the end. Find it's average velocity over the entire path.

- Relevant Equations
- ##v_{av-x}=\frac {Δx} {Δt}##

The car covers half of the road with an average velocity of v, so the elapsed time is equal to: ##t_1=\frac {d/2} {v}=\frac {d} {2v}##

And it covers 1/4 of the road with an average velocity of 2v, so the elapsed time is equal to: ##t_2=\frac {d/4} {2v}=\frac {d} {8v}##

Then it covers 1/8 of the road with an average velocity of 4v, so the elapsed time is equal to: ##t_3=\frac {d/8} {4v}=\frac {d} {32v}##

And until the end ...

##v_{av-x}= \frac {Δx} {Δt}=\frac {d/2+d/4+d/8+...} {d/2v+d/8v+d/32v+...}=\frac {v/2+v/4+v/8+...} {1/2+1/8+1/32+...}##

##v_{av-x}=v\frac {1/2+1/4+1/8+...} {1/2+1/8+1/32+...}=v\frac {1/2+1/8+1/32+...} {1/2+1/8+1/32+...}+v\frac {1/4+1/16+1/64+...} {1/2+1/8+1/32+...}=v+\frac {v} {2}(\frac {1/2+1/8+1/32+...} {1/2+1/8+1/32+...})=v+\frac {v} {2}=1.5v##

I think my answer is correct but i wanted to know if there is an easier answer.

And it covers 1/4 of the road with an average velocity of 2v, so the elapsed time is equal to: ##t_2=\frac {d/4} {2v}=\frac {d} {8v}##

Then it covers 1/8 of the road with an average velocity of 4v, so the elapsed time is equal to: ##t_3=\frac {d/8} {4v}=\frac {d} {32v}##

And until the end ...

##v_{av-x}= \frac {Δx} {Δt}=\frac {d/2+d/4+d/8+...} {d/2v+d/8v+d/32v+...}=\frac {v/2+v/4+v/8+...} {1/2+1/8+1/32+...}##

##v_{av-x}=v\frac {1/2+1/4+1/8+...} {1/2+1/8+1/32+...}=v\frac {1/2+1/8+1/32+...} {1/2+1/8+1/32+...}+v\frac {1/4+1/16+1/64+...} {1/2+1/8+1/32+...}=v+\frac {v} {2}(\frac {1/2+1/8+1/32+...} {1/2+1/8+1/32+...})=v+\frac {v} {2}=1.5v##

I think my answer is correct but i wanted to know if there is an easier answer.