a. ![x(t) = vt' = v(t+4) = 6(t+4) [m]](https://tex.z-dn.net/?f=x%28t%29%20%3D%20vt%27%20%3D%20v%28t%2B4%29%20%3D%206%28t%2B4%29%20%5Bm%5D)
Let's call:
the time calculated from the moment Diane starts her motion and
the time calculated from the moment Derek starts his motion. Derek starts his motion 4 seconds before Diane: this means that has an "advantage" of 4 seconds, so we can write

Then:
is the uniform speed of Derek
Derek is moving in a uniform motion, so Derek's position will be given by (assuming that the starting position is zero):
![x(t) = vt' = v(t+4) = 6(t+4) [m]](https://tex.z-dn.net/?f=x%28t%29%20%3D%20vt%27%20%3D%20v%28t%2B4%29%20%3D%206%28t%2B4%29%20%5Bm%5D)
We see that this is correct: in fact, when t=0 (instant when Diane starts her motion), Derek has already travelled for

b. ![x'(t)= \frac{1}{2}at^2 =\frac{1}{2}(2)t^2=t^2 [m]](https://tex.z-dn.net/?f=x%27%28t%29%3D%20%5Cfrac%7B1%7D%7B2%7Dat%5E2%20%3D%5Cfrac%7B1%7D%7B2%7D%282%29t%5E2%3Dt%5E2%20%5Bm%5D)
t is the time calculated from the moment Diane starts her motion. Diane is moving by accelerated motion, with constant acceleration
and initial velocity
, so its position at time t is given by the law of uniform accelerated motion:

c. t = 8.74 s
The time t at which Diane catches up with Derek is the time t at which the positions of the two persons is equal:

By solving, we have:

Which has two solutions:
t = -2.74 s --> negative, we can discarde it
t = 8.74 s --> this is our solution
d. 76.4 m
When Diane catches Derek, at t=8.74 s, she has covered the following distance:

We can verify that Derek is at the same position:
