Find the inverse

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. 

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