Well, let's say y is v(t). It'll make it simpler, or at least... it will for me. :)
Then the equation is now y=100(1-t/40)^2
To find the inverse you need to switch the y and x, or in this case, y and t.
That would make t=100(1-y/40)^2
Solve for y
First divide both sides by 100 to make t/100=(1-y/40)^2
Then square root both sides to get √(t/100)=1-y/40, which simplifies to √(t)/10=1-y/40.
Then subtract 1 to both sides to get -1+√(t)/10=-y/40.
Next, multiply -1 to both sides to get 1-√(t)/10=y/40
Then multiply 40 to both sides to get 40-4√(t)=y
That is y=40-4√(t)
Which is v(t)=40-4√(t) because I had used y instead of v(t).
And that's the answer
v(t)=40-4√(t)
Correct me if I'm wrong. 
), so I used a calculator.