Chapter 2, Problem 47P

(III) Mary and Sally are in a fool race (Fig. 2-43). When Mary is 22 m from the finish line, she has a speed of 4.0 m/s and is 5.0 m behind Sally, who has a speed of 5.0 m/s. Sally thinks she has an easy win and so. during the remaining portion of the race, decelerates at a constant rate of 0.50 m/s² to the finish line. What constant acceleration does Mary now need during the remaining portion of the race. If she wishes to cross the finish line side-by-side with Sally?

FIGURE 2-43 Problem 47.

47. For the runners to cross the finish line side-by-side means they must both reach the finish line in the same amount of time from their current positions. Take Mary's current location as the origin. Use Eq. 2-12b.

For Sally: 22 = 5 + 5t + $\frac{1}{2}$(−.5) → t² − 20t + 68 = 0 →

$t=\frac{20\pm \sqrt{{20}^2-4(68)}}{2}=4.343\text{s}.15.66\text{s}$

The first time is the time she first crosses the finish line, and so is the time to be used for the problem. Now find Mary's acceleration so that she crosses the finish line in that same amount of time.

For Mary: $22=0+4t+\frac{1}{2}a{t}^2\to a=\frac{22-4t}{\frac{1}{2}{t}^2}=\frac{22-4(4.343)}{\frac{1}{2}{(4.343)}^2}=0.49\text{m}/{\text{s}}^2$