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## Homework Statement

(A) A damped oscillator is described by the equation

m x′′ = −b x′− kx .

What is the condition for critical damping? Assume this condition is satisfied.

(B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0. Determine the position x(t) for t > 0.

(C) Suppose k/m = (2π rad/s)2 and v0=10 m/s. Plot, by hand, an accurate graph of x(t). Use graph paper. Use an appropriate range of t.

## Homework Equations

For critically damped, β

^{2}= w

_{0}

^{2}

where β = b/(2m) and w

_{0}= √(k/m)

## The Attempt at a Solution

Ok, for this problem, what I did initially was find the general form of position for a critically damped oscillator, which is:

x(t) = (A + B*t)*e

^{-β*t}

and the velocity function is:

v(t) = -Aβe

^{-βt}+ (Be

^{-βt}- Bβte

^{-βt})

Using the conditions given, I found:

x(0) = A (obviously) which we don't know x(0)

B = v

_{0}+ Aβ

and x(t) can be rewritten as:

x(t) = A(e

^{-βt}+ βte

^{-βt}) + v

_{0}te

^{-βt}

This is where I run into a wall. I can't seem to solve for A. I believe that x(0) should also be the max displacement since there is no driver for the impulse force, so A should be the max displacement, but this doesn't seem to get me anywhere. Any help on solving for A? I know how to do the rest other than that.