# Running / walking for half distance

## Homework Statement

Arthur and Robin both can run at speed v_r and walk at speed v_w, with v_r > v_w. They set off together on a journey of distance D. Robin walks half of the distance and runs the other half. Arthur walks half of the time and runs the other half.

How long does it take Robin to cover the distance D?

Express the time taken by Robin in terms of v_r, v_w, and D.

## Homework Equations

v = d/t and t = d/v

## The Attempt at a Solution

total time = (D/2)/(v_w) + (D/2)/(v_r)

= D/(2v_w) + D/(2v_r)

I just wanted to check before I submit this , thanks for any help

Last edited:

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kuruman
Homework Helper
Gold Member
Nope. If t = D/v, then D/(2vw) is the time it takes to cover distance D/2 with walking speed vw and likewise for the running time in your expression. What you have is Arthur's total time, not Robin's.

Thanks for the response I got the names mixed up and I edited my first post. Got the first part correct.

The next question is
"Find Robin's average speed for covering the distance D.
Express Robin's average speed in terms of vr and vw".

v_avg = D / [(D/2v_w) + (D/2v_r)]
= 1 / [(1/2v_w) + (1/2v_r)]

= 2v_w + 2v_r

that last simplification Im not sure about though.. Thanks for any more help

SammyS
Staff Emeritus
Homework Helper
Gold Member
Thanks for the response I got the names mixed up and I edited my first post. Got the first part correct.

The next question is
"Find Robin's average speed for covering the distance D.
Express Robin's average speed in terms of vr and vw".

v_avg = D / [(D/2v_w) + (D/2v_r)]
= 1 / [(1/2v_w) + (1/2v_r)]
The above is correct.

The following line is not correct.
= 2v_w + 2v_r

that last simplification Im not sure about though.. Thanks for any more help
Multiply the numerator and denominator of $\displaystyle \ \frac {1}{\displaystyle \frac{1} { 2v_w }+\frac{1} { 2v_r } } \$ by $\displaystyle \ 2 v_w v_r \,.$