Moving plane

Define the position and velocity of the block, given that the ramp can move ( $\theta$ is the ramp angle with the horizontal).

$\large{x_2 = x_1 + \alpha \, cos \theta \\ y_2 = \alpha \, sin \theta \\ \dot{x_2} = \dot{x_1} + \dot{\alpha} \, cos \theta \\ \dot{y_2} = \dot{\alpha} \, sin \theta \\ v_2^2 = \dot{x_2}^2 + \dot{y_2}^2 = \dot{x_1}^2 + 2 \, \dot{x_1} \, \dot{\alpha} \, cos \theta + \dot{\alpha}^2 }$

Kinetic Energy:

$\large{E = \frac{1}{2} m_1 \, \dot{x_1}^2 + \frac{1}{2} m_2 \, (\dot{x_1}^2 + 2 \, \dot{x_1} \, \dot{\alpha} \, cos \theta + \dot{\alpha}^2)}$

Potential Energy:

$\large{U = m_2 \, g \, \alpha \, sin \theta}$

Lagrangian:

$\large{L = E - U = \frac{1}{2} m_1 \, \dot{x_1}^2 + \frac{1}{2} m_2 \, (\dot{x_1}^2 + 2 \, \dot{x_1} \, \dot{\alpha} \, cos \theta + \dot{\alpha}^2) - m_2 \, g \, \alpha \, sin \theta}$

Equations of Motion:

$\large{\frac{d}{dt} \frac{\partial{L}}{\partial{\dot{x_1}}} = \frac{\partial{L}}{\partial{x_1}} \\ \frac{d}{dt} \frac{\partial{L}}{\partial{\dot{\alpha}}} = \frac{\partial{L}}{\partial{\alpha}}}$

Evaluating the equations of motion yields:

$\large{(m_1 + m_2) \, \ddot{x_1} + m_2 \, cos \theta \, \ddot{\alpha} = 0 \\ m_2 \, cos \theta \, \ddot{x_1} + m_2 \, \ddot{\alpha} = -m_2 \, g \, sin \theta}$

Substituting in for the $\ddot{\alpha}$ term yields:

$\large{\ddot{x_1} = \frac{m_2 \, g \, sin \theta \, cos \theta}{m_1 + m_2 - m_2 \, cos^2 \theta}}$

Plugging in numbers (assuming $g = 10$ ):

$\large{\ddot{x_1} = \frac{(1) \, (10) \, (1/2)}{10 + 1 - (1) \, (1/2)} \approx 0.47619 }$