We can solve this problem by treating the three blocks as a system and applying Newton's Second Law of Motion to analyze the forces.

Step 1: Calculate the total acceleration of the system

The total mass of the system is the sum of the masses of the blocks: $m_{\text{total}} = m_1 + m_2 + m_3 = 1.0\,\text{kg} + 2.0\,\text{kg} + 4.0\,\text{kg} = 7.0\,\text{kg}$

The total force applied to the system is 14 N, so using Newton's Second Law $F = ma$, we can calculate the acceleration $a$ of the system: $a = \frac{F_{\text{total}}}{m_{\text{total}}} = \frac{14\,\text{N}}{7.0\,\text{kg}} = 2.0\,\text{m/s}^2$

Step 2: Calculate the force between the 2.0 kg and 4.0 kg blocks

Now, we focus on the 4.0 kg block. The force that causes the 4.0 kg block to accelerate comes from the 2.0 kg block. Using Newton's Second Law again for the 4.0 kg block, we can calculate this force as: $F_{2 \rightarrow 3} = m_3 \cdot a = 4.0\,\text{kg} \cdot 2.0\,\text{m/s}^2 = 8.0\,\text{N}$

Thus, the force that the 2.0 kg block exerts on the 4.0 kg block is 8.0 N.

Would you like a more detailed explanation or have any additional questions?

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Here are five related questions to further explore:

1. What is the force exerted by the 1.0 kg block on the 2.0 kg block?
2. How would the system behave if friction were introduced between the table and the blocks?
3. What happens to the acceleration if a larger force is applied to the 1.0 kg block?
4. How would the forces between the blocks change if the masses of the blocks were different?
5. Can you determine the force exerted on the 1.0 kg block by the 2.0 kg block?

Tip: Newton's Third Law states that forces between two objects are equal and opposite. Use this law when analyzing interactions between blocks!