a) We will determine the distance traveled by each object first, that is:
Object 1:
[tex]d_1=(9m/s)(4s)+\frac{1}{2}(3m/s^2)(4s)^2\Rightarrow d_1=60m[/tex]Object 2:
[tex]d_2=(0m/s)(4s)+\frac{1}{2}(5m/s^2)(4s)^2\Rightarrow d_2=40m[/tex]So, both objects are 20 meters apart after 4 seconds.
b) Now, we determine the distance when they have the same velocity, that is:
First, we determine the time it takes for them to have the same velocity:
[tex]\frac{1}{2}(5m/s^2)t^2=(9m/s)t+\frac{1}{2}(3m/s^2)t^2\Rightarrow(1m/s^2)t^2-(9m/s)t=0[/tex][tex]\Rightarrow t=\frac{-(-9)\pm\sqrt[]{(-9)^2-4(1)(0)}}{2(1)}\Rightarrow\begin{cases}t=9s \\ \\ t=0s\end{cases}[/tex]So, after 9 seconds they will have the same velocity, now we determine the distance between them:
Object 1:
[tex]d_1=(9m/s)(9s)+\frac{1}{2}(3m/s^2)(9s)^2\Rightarrow d_1=202.5m[/tex]Object 2:
[tex]d_2=(0m/s)(9s)+\frac{1}{2}(5m/s^2)(9s)^2\Rightarrow d_2=202.5m[/tex]The distance between the two objects when they have the same velocity is 0 meters. They are in the same place.
