two runners are competing in a 400-meter dash. The first runs at 6m/s and maintains this speed. The second runner starts at 10m/s and decelerates at 1.8m/s^2. Who will finish first? What will be their time difference?

Answer 1

d = Vi t + .5 a t^2

runner 1
400 = 6 t and a = 0
t = 400/6

runner 2
400 = 10t - 0.9 t^2
.9 t^2 - 10 t + 400 = 0 solve quadratic
t = complex number, never makes it :)
The velocity turns negative before the runner reaches the finish
Let's see when v = 0
v = 10 -1.8 t
v = 0 when t = 10/1.8 = 5.6 seconds
then
d = 10 (5.6) - 0.9 (5.6)^2 = 56 - 28.2 = 27.8
so this second runner starts backing up after 28 meters :)