Mechanics, PE to position function

[tjkubo]

Homework Statement

A 3 kg object is moving along the x-axis where U(x) = 4x². At x = -.5, v = +2. Find the object's position and KE as functions of time. Assume x = 0 at time t = 0. All forces acting on the object are conservative.

Homework Equations

ME = U + K
K = (1/2)mv²
F = dU/dx
F = ma

The Attempt at a Solution

Using initial conditions, ME = 4.
F = dU/dx = 8x
F = ma
8x = (3)d²x/dt²
This is where I got stuck. I was attempting to solve for x(t), find v(t), then use that to find K(t). Assuming everything else is correct, how do you solve a second order differential equation like this? Otherwise, please correct me.

 

[rock.freak667]

Just solve

$$\frac{d^2x}{dt^2}- \frac{8}{3}x=0$$

Do you know how to solve a second order differential equation with constant coefficients?


EDIT: http://www.sosmath.com/diffeq/second/constantcof/constantcof.html"

 

[tjkubo]

I am still having trouble solving for x(t).
I got $$x=c_1e^{\sqrt{\frac{8}{3}}t}+c_2e^{-\sqrt{\frac{8}{3}}t}$$
and $$c_1+c_2=0$$
but since there is no initial value associating time and velocity, I can't find the constants.

 

[rock.freak667]

I am still having trouble solving for x(t).
I got $$x=c_1e^{\sqrt{\frac{8}{3}}t}+c_2e^{-\sqrt{\frac{8}{3}}t}$$
and $$c_1+c_2=0$$
but since there is no initial value associating time and velocity, I can't find the constants.

How did you get ME=4 by chance?

Also F=-dU/dx not F=+dU/dx

 

[Delphi51]

Can you use the "At x = -.5, v = +2" condition?
Would it work to begin with
ME = U + K
4 = 4x^2 + 1/2*mv^2 (which includes the x = -.5 condition)
4 = 4x^2 + 1.5(dx/dt)^2
The solution to this differential equation would have only one constant, which you should be able to get using the x=0 at t=0 condition.

 

[tjkubo]

How did you get ME=4 by chance?

Also F=-dU/dx not F=+dU/dx

My bad, ME = 7. I forgot to square. (Is is correct to assume that ME is constant?)
Anyway, when I retried solving the differential equation with the initial conditions, I ended up getting 0 = 0 while solving for the constants. ?

 

[tjkubo]

4 = 4x^2 + 1.5(dx/dt)^2
The solution to this differential equation would have only one constant, which you should be able to get using the x=0 at t=0 condition.

How would you solve this differential equation? The (dx/dt)^2 term throws me off.

 

[Delphi51]

dx/dt = sqrt(2/3)*sqrt(4 - 4x^2)
sqrt(2/3) dt = dx/sqrt(4 - 4x^2)
Integrate both sides. Doesn't look bad - trig substitution if I'm not mistaken.

 

[tjkubo]

Yes! I got
$$x=\frac{\sqrt{7}}{2}\sin{\sqrt{\frac{4}{3}}\,t}$$
That was much simpler than what I was doing.
Thanks Delphi!