For instance, one of the kinematic equations is :
$$v_f^2 = v_i^2 + 2ad$$
where $v_f$ is final velocity, $v_i$ is initial velocity, $a$ is acceleration, and $d$ is displacement.
Say for instance a guy rides a bike in circles for hours with initial velocity of $10\ m/s$ and has an acceleration of $1\ m/s^2$, but he finished on the same spot he started on. His displacement would be $0$, right?
So the $2ad$ part of the equation would turn out to be $2 \times 1 \times 0 = 0$.
This means we're left with
$$v_f^2 = v_i^2 + 0.$$
Taking the square root of each side means the final velocity equals the initial velocity, but I stated that he sped up in the problem and thus the final velocity should be higher.
Why does this not work? Do I have to use distance instead of displacement for an equation like this?