1. The problem statement, all variables and given/known data All right, the problem is that I have a motor that's acceleration at a rate of a = 6 theta rad/s^2. For the bigger problem that this is part of, I need to find out the velocity function. How do I do this if a isn't given in terms of t? 2. Relevant equations omega = d theta/dt a = d omega / dt 3. The attempt at a solution All I can think of is that a = 6 theta integrate both sides for t v = 6 theta t theta = 3 theta t^2? that doesn't make sense! and theta is really theta(t), so how can I integrate that function if I don't know what i is? help me calculus gurus!
I Think you have to use the result [tex]a=x''=\frac{d}{dx} (\frac{1}{2}v^2)[/tex]. That result can be obtained as such - [tex]\frac{d}{dx} ((1/2) v^2) = \frac{d}{dv} ((1/2) v^2) \cdot \frac{dv}{dx}=v\frac{dv}{dx}[/tex] by the chain rule. Another application of the chain rule: [tex]v\frac{dv}{dx} = \frac{dx}{dt}\cdot\frac{dv}{dx} = \frac{dv}{dt}=a[/tex]
actually as it turned out: a(t) = 6 theta better put as a(t) = 6 theta(t) a(t) is just the second derative of theta(t), so theta''(t) = 6 theta(t) tossing away the 6 mentally for a second, what function equals its own second derative? e does! so let's play with that and see what we get if theta(t) = e^t, then theta''(t) = e^t. Okay, but we need to bring that 6 back in. So, theta (t) = e^sqrt6 t, theta'(t) = v(t) = sqrt6 e^sqrt6 t, and v''(t) = a(t) = 6 e^sqrt6 t. To check it:a(t) = 6 e^sqrt6 t. e^sqrt6 t = theta(t) substitute and we get back to the original a(t) = 6 theta. yay :) using this I got the right answer for my problem! But I'm on another problem (last one, promise) with similar premises. omega(t) = 5 (theta(t))^2 is what I'm given. So theta'(t) = 5 (theta(t))^2 Toss away the 5, we can deal with it later...but what function equals its derative when you square it?