To solve this problem, we start with the given parameters:

• Mass, $m = 1$
• Damping constant, $c = 10$
• Spring constant, $k = 25$
• Initial position, $x_0 = 4$
• Initial velocity, $v_0 = -5$

Step 1: Determine the Type of Damping

First, we find the damping ratio $\zeta$ using the formula:

$\zeta = \frac{c}{2\sqrt{mk}}$

Substituting the values:

$\zeta = \frac{10}{2\sqrt{1 \cdot 25}} = \frac{10}{10} = 1$

Since $\zeta = 1$, the system is critically damped.

Step 2: Position Function $x(t)$

For a critically damped system, the position function can be expressed as:

$x(t) = (C_1 + C_2 t)e^{-\zeta \omega_0 t}$

where $\omega_0 = \sqrt{\frac{k}{m}}$.

Calculating $\omega_0$:

$\omega_0 = \sqrt{\frac{25}{1}} = 5$

Thus, $x(t)$ becomes:

$x(t) = (C_1 + C_2 t)e^{-5t}$

Step 3: Finding Constants $C_1$ and $C_2$

Using the initial conditions to find $C_1$ and $C_2$:

1. At $t = 0$:

$x(0) = C_1 = 4$

2. Velocity is the derivative of position:

$v(t) = \frac{dx}{dt} = \frac{d}{dt}[(C_1 + C_2 t)e^{-5t}]$

Using the product rule:

$v(t) = (C_2 e^{-5t} + (C_1 + C_2 t)(-5)e^{-5t}) = e^{-5t}(C_2 - 5C_1 - 5C_2 t)$

At $t = 0$:

$v(0) = C_2 - 5C_1 = -5 \implies C_2 - 20 = -5 \implies C_2 = 15$

Thus, the position function is:

$x(t) = (4 + 15t)e^{-5t}$

Step 4: Undamped Position Function $u(t)$

For the undamped system (where $c = 0$), the position function is:

$u(t) = C_0 \cos(\omega_0 t - \alpha_0)$

Using $x(0) = 4$ and $v(0) = -5$:

1. At $t = 0$:

$u(0) = C_0 \cos(-\alpha_0) = C_0 \implies C_0 = 4$

2. Velocity is:

$v(t) = -C_0 \omega_0 \sin(\omega_0 t - \alpha_0)$

At $t = 0$:

$v(0) = -C_0 \omega_0 \sin(-\alpha_0) = -4 \cdot 5 \sin(-\alpha_0) = -20 \sin(-\alpha_0)$

Setting $v(0) = -5$:

$-20 \sin(-\alpha_0) = -5 \implies \sin(-\alpha_0) = \frac{1}{4} \implies \alpha_0 = -\arcsin\left(\frac{1}{4}\right) \approx -0.2527$

Thus, the undamped position function is:

$u(t) = 4 \cos(5t + 0.2527)$

Summary of Results

1. Position function $x(t) = (4 + 15t)e^{-5t}$ (critically damped).
2. Undamped position $u(t) = 4 \cos(5t + 0.2527)$.

Step 5: Graph Comparison

To illustrate the effect of damping, we would plot both $x(t)$ and $u(t)$ on the same graph.

If you need the graph created, I can assist with that as well. Do you have any questions or need further details on any step? Here are a few follow-up questions:

1. Would you like to see the graphical representation of the functions?
2. Do you want a deeper explanation of the damping types?
3. Would you like to learn how to derive these formulas step-by-step?
4. Are you interested in examples of real-world applications of these concepts?
5. Do you need assistance with related differential equations?

Tip: When analyzing damping in physical systems, remember that the damping ratio $\zeta$ plays a crucial role in determining the behavior of the system, guiding the design of systems like car suspensions or mechanical watches.