# Running and Walking Problem| University Physics

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

ANSWER: $t_R=\frac{D}{2v_rv_w}\left(v_w+v_r\right)$

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 $v_w$.

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

Now find the time Rick spends running.

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

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

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

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

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

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.

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

ANSWER: $t_T=\frac{2D}{v_w+v_r}$

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

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

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

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

### PART D. Who covers the distance D more quickly?

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?

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

This is just simple subtraction between the two computed times.