Answer:
[tex]x(t)=-15t^2+5t^3+12[/tex]
Explanation:
To find the expression for the position x of the particle as a function of time t you need to integrate the expression for acceleration two times. To find the integral constants you are already given the initial conditions for both the position and the velocity which are:
[tex]x(t=0)=12\\v(t=0)=0[/tex]
The first integration gives you the velocity:
[tex]v(t)=\int a(t) dt=-30t+15t^2+C_1[/tex]
You get the constant by using the second initial condition:
[tex]C_1=0[/tex]
To get the expression for the position you need to integrate again:
[tex]x(t)=\int v(t)dt=-15t^2+5t^3+C_2[/tex]
where [tex]C_2=12\\[/tex] using the first initial condition.
You can now plot the graph. To get the total distance travelled you can integrate the expression for the velocity twice and sum the two integrals (from 0 to 1 and from 1 to 3):
[tex]\int_0^1|v(t)|dt+\int_1^3|v(t)|dt=[/tex]20 m
