Non-constant acceleration kinematics

[blackrose75]

Homework Statement

The acceleration of a marble in a certain fluid is proportional to the speed of the marble squared, and is given (in SI units) by a = -3.00v^2 for v > 0. If the marble enters this fluid with a speed of 1.50 m/s, how long will it take before the marble's speed is reduced to half of its initial value?


2. The attempt at a solution
I know that this question appeared here a while ago (found an archive to this exact one from two years ago) but I wasn't too sure with some of the help they gave so I'm asking again if someone would kindly help. I know that it involves a=dv/dt so that I end up with something like (v2-v1)=-3v^2(t-0). That may be wrong, I'm in a high school AP class and am just starting with some basic derivations in my calc class, so it is very possible that there's a flaw in all that (and in some of my terminology or whatnot) but if I'm on the right track my main concern would be the v^2. I saw on another post that someone did

dv/v^2 = -kdt

-1/v = -kt +C

but I'm not sure how they got there (besides it looking like they took the integral)
Again, things may be wrong and I wouldn't be surprised, but any help would be appreciated, thanks.

 

[LowlyPion]

Homework Statement

The acceleration of a marble in a certain fluid is proportional to the speed of the marble squared, and is given (in SI units) by a = -3.00v^2 for v > 0. If the marble enters this fluid with a speed of 1.50 m/s, how long will it take before the marble's speed is reduced to half of its initial value?

2. The attempt at a solution
I know that this question appeared here a while ago (found an archive to this exact one from two years ago) but I wasn't too sure with some of the help they gave so I'm asking again if someone would kindly help. I know that it involves a=dv/dt so that I end up with something like (v2-v1)=-3v^2(t-0). That may be wrong, I'm in a high school AP class and am just starting with some basic derivations in my calc class, so it is very possible that there's a flaw in all that (and in some of my terminology or whatnot) but if I'm on the right track my main concern would be the v^2. I saw on another post that someone did

dv/v^2 = -kdt

-1/v = -kt +C

but I'm not sure how they got there (besides it looking like they took the integral)
Again, things may be wrong and I wouldn't be surprised, but any help would be appreciated, thanks.

Welcome to PF.

The solution you are referencing is using acceleration as dv/dt. Since what you are looking for is a value for the difference in time that should be found by employing the difference in velocity of the given problem. You separate the V terms to the V side. (This eliminates I think your main concern.) And then integrate both sides.

That yields the relationship V to dv and t to dt.

-1/V = -k/t + C

Now using this relationship, subtract from one known Vo the other known V1, and with the initial t being 0 you can solve for t1

(-1/1.5) - (-1/ .75) = (-3*0+c) - (-3t1+c) = 3*t1

That's at least how I see the problem.

 

[blackrose75]

Thanks for the help, but I'm sorry to say I'm still a little bit lost on that...