#47, the whole premise is manipulating equations algebraically but idk how to manipulate this

Just exactly like you did for #46, and like you would do with any other equation . . . Do whatever is necessary to get ' a ' all by itself on one side of the equation, but whatever you do to one side must also immediately be done to the other side.

Here's a set of steps that will untangle this one:

You said that . . . vf² = v0² + 2a·(Xf-X0)

Subtract v0² from each side . . . vf² - v0² = 2a·(Xf-X0)

Divide each side by 2·(Xf-X0) . . . (vf²-v0²)/[2·(Xf-X0)] = a

if the thermos has a vacuum innner, then heat through conv currents and conduction are reduced = vacuum flask