Bowling ball dropped into water, find velocity as it strikes the bottom

[bbraves7]

Homework Statement

A bowling ball is dropped from a boat so that it strikes the surface of a lake with a speed of 25 ft/s. Assuming the ball experiences a downward acceleration of a=10-.9v² when in the water, determine the velocity of the ball when it strikes the bottom of the lake.

Homework Equations

v₀=25 ft/s
da/dt=v
dv/dt=y

The Attempt at a Solution

I've tried this problem a few different ways, the first by substituting v into the given a=10-.9v² and then integrating, but that does not work because v₀ is the initial velocity and not the velocity as it travels through the water, which obviously will change based on the acceleration. I think I am missing that the ball will slow down as it is in the water, and thus the acceleration will slow down according to the velocity at that time. I just can't seem to put that into the form of any equations I'm familiar with. I have the solutions manual for this textbook, but the way they solve it they include an equation a=10-.9v²=k(c²-v²) but I don't know where they pulled that from because I can't find anything close to that in the text. I desperately want to comprehend this problem rather than just regurgitate the solution. Any help will be greatly appreciated.

 

[lewando]

Some thoughts:

da/dt=v
dv/dt=y

You probably meant dv/dt=a, dy/dt=v.

You are given V₀ and a, but not the depth of the lake, which should be specified.

a=10-.9v², has units in m/s²? ft/s²?

I think the intent of specifying the acceleration that way is to indicate a terminal velocity (when a = 0). Though, if this is a very shallow lake, it may not reach terminal.

 

[bbraves7]

Sorry, it was very late last night when I posted this problem.

I meant to put dy/dt=v, dv/dt=a.

The depth of the lake is 30m. v₀= 25ft/s.

Sorry again about that. I had been working on this for about two hours so my brain was fried.