Running and Walking Problem| University Physics

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

PART A. How long does it take Rick to cover the distance D?


Find the time that it takes Rick to walk the first half of the distance, that is, to travel a distance D/2 at speed \displaystyle v_w.

\displaystyle t_{w,R}=\frac{D}{2v_w}

Now find the time Rick spends running.

\displaystyle t_{r,R}=\frac{D}{2v_r}

Now just add the two times up and you’re done.

\displaystyle t_R=\frac{D}{2v_w}+\frac{D}{2v_r}=\frac{D}{2v_wv_r}\left(v_w+v_r\right)

PART B. Find Rick’s average speed for covering the distance D.


You were given the total distance and have calculated the total time. Recall that average speed is equal to the total distance traveled divided by the amount of time it took to travel this distance.

\displaystyle v_{ave,\:R}=\frac{2v_rv_w}{v_w+v_r}

PART C. How long does it take Tim to cover the distance?


Tim walks at speed \displaystyle v_w half the time and runs at speed \displaystyle v_r for the other half.

\displaystyle v_{ave,\:T}=\frac{v_w+v_r}{2}

The time is just the distance divided by the average speed.

\displaystyle t_T=\frac{D}{\frac{v_w+v_r}{2}}=\frac{2D}{v_r+v_w}

PART D. Who covers the distance D more quickly?

Answer: Tim

Imagine that both Rick and Tim do all of their walking before they start to run. Rick will start running when he has covered half of the total distance. When Tim reaches half of the total distance, will he already have started running?

PART E. In terms of given quantities, by what amount of time, Δt, does Tim beat Rick?


\displaystyle \Delta t=\frac{D\left(v_w-v_r\right)^2}{2v_rv_w\left(v_r-v_w\right)}

This is just simple subtraction between the two computed times.

PART F. In the special case that vr=vw, what would be Tim’s margin of victory Δt(vr=vw)?


If vr=vw, is the any difference between what Tim and Rick do?