Simple math problem in one dimension

[Slusho]

Homework Statement

A particle of mass m is subjected to a net force F(t) given by F(t)=F₀(1-t/T)i; that is F(t) equals F₀ at t=0 and decreases linearly to zero in time T. The particle passes the origin x=0 with velocity v₀i. Show that at the instant t=T and F(t) vanishes, the speed v and distance x traveled are given by v(T)=v₀+a₀T/2, and x(T)=v₀T+a₀T²/3, where a₀=F₀/m is the initial acceleration. Compare these results with Eqs (v_x=v_0x+a_xt and x=x₀+v_0xt+1/2a_xt²).

Homework Equations

I should only need those listed in the problem

The Attempt at a Solution

F=ma=F₀(1-t/T)
F₀=ma₀
a=a₀(1-t/T)
a=a₀-a₀t/T

v=v₀+at
v=v₀+(a₀-a₀t/T)t

When t=T, those a₀'s cancel each other out and you end up with zero. I've also tried starting with the other one, and taking derivatives, to no avail. I can't get the math to work out to give me those coefficients.

Help is GREATLY appreciated.

 

[E_M_C]

So you've already written Newton's 2nd Law, that's important. But what are the time derivative relationships between acceleration, velocity, and position? Use those relationships to work backwards from Newton's 2nd Law to obtain v(t) and x(t).