1. Initial State and Collision

Equilibrium of Plate A ($m$):
$k x_0 = mg \implies k = mg/x_0$. (Equilibrium is at compression $x_0$).

Collision:
Plate B ($2m$) falls from height $3x_0$. Velocity $v = \sqrt{6gx_0}$.
They move together (mass $3m$). By momentum conservation:
$2m v = 3m v’ \implies v’ = \frac{2}{3}\sqrt{6gx_0}$.

2. Oscillation Analysis

New Parameters:
Total mass $M_{total} = 3m$.
New Mean Position: $k x_{mean} = 3mg \implies x_{mean} = 3x_0$ (compression).
Angular frequency: $\omega^2 = \frac{k}{3m} = \frac{mg/x_0}{3m} = \frac{g}{3x_0}$.

State at Impact:
Position is $x_0$ (compression). This is a displacement $y = x_0 – 3x_0 = -2x_0$ from the new mean position (taking downward as positive).
Velocity $v’ = \frac{2}{3}\sqrt{6gx_0}$.

Amplitude ($A$):
Using $v^2 = \omega^2 (A^2 – y^2)$:
$\frac{4}{9}(6gx_0) = \frac{g}{3x_0} (A^2 – (-2x_0)^2)$
$\frac{8}{3}gx_0 = \frac{g}{3x_0} (A^2 – 4x_0^2)$
$8x_0^2 = A^2 – 4x_0^2 \implies A^2 = 12x_0^2 \implies A = 2\sqrt{3}x_0 \approx 3.46 x_0$.

3. Separation at Natural Length

The plates are not attached. Separation occurs when the normal force between them becomes zero. Since Plate B is only affected by gravity and N, if N=0, B accelerates at $g$. If the spring pulls Plate A down with acceleration $> g$, they separate. This happens as soon as the spring enters the extension phase (passes Natural Length).

Natural Length Position:
Natural Length corresponds to compression $0$. This is $3x_0$ above the mean position ($y = -3x_0$).
Since Amplitude $A \approx 3.46x_0$, the system reaches Natural Length.

Velocity at Separation ($v_{sep}$):
At Natural Length ($y = -3x_0$):
$v_{sep}^2 = \omega^2 (A^2 – (3x_0)^2) = \frac{g}{3x_0} (12x_0^2 – 9x_0^2) = \frac{g}{3x_0}(3x_0^2) = gx_0$.
$v_{sep} = \sqrt{gx_0}$ (upwards).

4. Maximum Height Calculation

After separation at Natural Length, Plate B moves as a free projectile under gravity.
Extra height gained: $h_{extra} = \frac{v_{sep}^2}{2g} = \frac{gx_0}{2g} = \frac{x_0}{2}$.

Total Height above Initial Position:
Initial position of A was at equilibrium ($x_0$ compression).
Separation point (Natural Length) is at distance $x_0$ above the initial position.
Total Max Height $H = (\text{Height to Separation}) + h_{extra}$
$H = x_0 + \frac{x_0}{2} = \frac{3x_0}{2}$.