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## Homework Statement

A weight of 8 pounds extends a spring 2 feet. It's assumed that the damping force that acts on the system is equal (number-wise) to alpha times the speed of the weight.

Determine the value of alpha > zero so x(t) is critically damped.

Determine x(t) if the weight is liberated from it's equilibrium position with a descending speed of 3 feet per second

Graph x(t)

## Homework Equations

I know i need to use Hooke's and then Newton's law

## The Attempt at a Solution

I'm fully aware of how to solve a second order differential equation.

I also know that, in the end, i have to achieve

x´´(t) + 4ax´(t)+16x(t)=0 ///

x(0)=0

x´(0)=3

Since that second order ODE has no g(x) i just use the homogeneous formula with the initial conditions to create the function i need.

I also know alpha needs to be 2 so this is a critically damped system because, here, the frequency squared is 16 and 2 times the number alongside x´(t) needs to be equal to the frequency (2a=w) => a=2

I found this problem in my DE course at university and i couldn't solve it because i couldn't even get to the problem, i am now trying to find sense to it but i need some directions

Thanks to anyone who made time to read this :D