Changing the Variable of Integration

[AxeluteZero]

Homework Statement

Suppose the acceleration of a particle is a function of x, where a_x(x) = (2.0 s_-2)x.

(a) If the velocity is zero when x = 1.0 m, what is the speed when x = 2.7 m?


(b) How long does it take the particle to travel from x = 1.0 m to x = 2.7 m?

Homework Equations

a = integral v dt = integral (integral (x)) dt

The Attempt at a Solution

This CAN be solved as a differential equation, but we haven't done those in my Calc course yet, so I have no idea how to solve it that way.

On the other hand, I know the problem is that acceleration is a function of x, hence a(x), and that it needs to be a function of time in order to change it over to velocity and then displacement (if needed). So, I tried figuring that out and go to this point:

a = $$\frac{dv}{dt}$$ = ($$\frac{dv}{dx}$$ * $$\frac{dx}{dt}$$)

$$\int\frac{dv}{dx}$$ = $$\int2x$$$$\frac{dx}{dt}$$

v = x² + c

Do I have this set up correctly? And, if so, wouldn't the integral (and thus the velocity function) end up being x² + some constant? And would that constant be related to part b, or inherited from the given info?

 

[rock.freak667]

Well you have that

a= (dv/dx)(dx/dt) and dx/dt is v.

So really you have that

v(dv/dx)=2x

or v dv = 2x dx

so integrate it now.

Then yes, you must use the conditions given in part a to get the constant of integration.