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
Correct me if I'm wrong.
Substituting x for t, this function is not one-to-one....thus....it has no inverse unless we restrict its domain
y = 100 ( 1 - t/40 )^2 divide both sides by 100
y / 100 = ( 1 - t/40)^2 take both roots of sides
±√ (y/100) = 1 - t/40 multiply both sides by -1
±√ ( y / 100 ) = t / 40 - 1 add 1 to both sides
1 ± √ ( y / 100 ) = t/40 multiply both sides by 40
40 ± 40 √ ( y / 100 ) = t swap t and y
40 ± 40 √ ( t / 100 ) = y for y, write f-1(t)
40 ± 40 √ ( t / 100 ) = f-1(t)
Here is the graph : https://www.desmos.com/calculator/lf5zoe7lei
If we restrict the original domain to ( -infinity, 40), the inverse is represented by the orange graph
If we restrict the original domain to (40, infinity), the inverse is represented by the red graph