To solve this problem, we will approach it in steps. Since the collision is completely elastic, we will use the laws of conservation of energy and momentum. The steps are as follows:

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Step 1: Velocity of each body before the collision

Using the conservation of mechanical energy, the potential energy of each body at the top of the incline is converted into kinetic energy just before the collision.

For body $m_1 = 1 \, \text{kg}$:

$m_1 g h_1 = \frac{1}{2} m_1 v_1^2$ $1 \cdot 9.81 \cdot 2 = \frac{1}{2} \cdot 1 \cdot v_1^2$ $v_1^2 = 2 \cdot 9.81 \cdot 2 = 39.24 \quad \Rightarrow \quad v_1 = \sqrt{39.24} \approx 6.26 \, \text{m/s}$

For body $m_z = 4 \, \text{kg}$:

$m_z g h_z = \frac{1}{2} m_z v_z^2$ $4 \cdot 9.81 \cdot 1 = \frac{1}{2} \cdot 4 \cdot v_z^2$ $v_z^2 = 2 \cdot 9.81 \cdot 1 = 19.62 \quad \Rightarrow \quad v_z = \sqrt{19.62} \approx 4.43 \, \text{m/s}$

So, just before the collision: $v_1 = 6.26 \, \text{m/s}, \quad v_z = 4.43 \, \text{m/s}$

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Step 2: Velocities after the elastic collision

For a completely elastic collision, both momentum and kinetic energy are conserved. The final velocities $v_1'$ and $v_z'$ can be determined using the following formulas:

$v_1' = \frac{(m_1 - m_z)v_1 + 2 m_z v_z}{m_1 + m_z}$ $v_z' = \frac{(m_z - m_1)v_z + 2 m_1 v_1}{m_1 + m_z}$

Substituting the values:

$v_1' = \frac{(1 - 4)(6.26) + 2(4)(4.43)}{1 + 4}$ $v_1' = \frac{-3 \cdot 6.26 + 2 \cdot 4 \cdot 4.43}{5}$ $v_1' = \frac{-18.78 + 35.44}{5} = \frac{16.66}{5} \approx 3.33 \, \text{m/s}$

$v_z' = \frac{(4 - 1)(4.43) + 2(1)(6.26)}{1 + 4}$ $v_z' = \frac{3 \cdot 4.43 + 2 \cdot 6.26}{5}$ $v_z' = \frac{13.29 + 12.52}{5} = \frac{25.81}{5} \approx 5.16 \, \text{m/s}$

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Final Velocities

After the collision: $v_1' \approx 3.33 \, \text{m/s}, \quad v_z' \approx 5.16 \, \text{m/s}$

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Would you like a detailed breakdown of any step or help with related concepts? Here are some related questions for further exploration:

1. How would the velocities change if the collision were perfectly inelastic?
2. What happens if friction on the incline is not negligible?
3. How does the angle of the incline affect the velocities before the collision?
4. What if the bodies were initially at different heights, e.g., $h_1 = 3 \, \text{m}$ and $h_z = 2 \, \text{m}$?
5. How would rotational motion influence the outcome, assuming the objects are spheres?

Tip: When solving elastic collision problems, always start by checking for energy and momentum conservation laws, as they are your key tools!