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To solve this problem, we’ll go through the following steps:

1. **Calculate the Total Mass**:
   The two boxes are a 1 kg box and a 3 kg box. Together, the total mass \( m_{total} \) is:
   \[
   m_{total} = 1 \, \text{kg} + 3 \, \text{kg} = 4 \, \text{kg}
   \]

2. **Determine the Acceleration**:
   The only horizontal force applied is 10 N on the 1 kg box, and since there is no friction, this force accelerates both boxes together. Using Newton's second law, \( F = ma \), we can solve for the acceleration \( a \) of the system:
   \[
   a = \frac{F}{m_{total}} = \frac{10 \, \text{N}}{4 \, \text{kg}} = 2.5 \, \text{m/s}^2
   \]

3. **Find the Contact Force between the Boxes**:
   Now, we need to determine the contact force between the two boxes, which is the force that the 3 kg box exerts on the 1 kg box (or vice versa, due to Newton's third law).

   To find this contact force, we can focus on the 3 kg box alone. Since it's accelerating at 2.5 m/s² (the same as the entire system), we apply Newton's second law to the 3 kg box:
   \[
   F_{contact} = m_{3 \, \text{kg}} \times a = 3 \, \text{kg} \times 2.5 \, \text{m/s}^2 = 7.5 \, \text{N}
   \]

Therefore, the contact force between the two boxes is **7.5 N**.

### Answer
The correct answer is:
- \( \boxed{7.5 \, \text{N}} \)

Would you like more details on any part of this solution, or have any questions?

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### Related Questions

1. What would happen to the acceleration if there was friction acting on both boxes?
2. How would the contact force change if the applied force were increased to 15 N?
3. If the 10 N force were applied to the 3 kg box instead, would the contact force be the same?
4. What is the role of Newton’s third law in determining the contact force?
5. How would we find the contact force if the masses of the boxes were unknown?

### Tip
When solving problems involving multiple objects in contact, treat the whole system to find the acceleration first, then analyze individual objects to determine internal forces.

A 1-kg and a 3-kg box are touching each other. A 10-N horizontal force is applied horizontally to the 1-kg box in order to accelerate both boxes across the level floor. Ignore friction forces and determine the acceleration of the boxes and the force acting between the boxes. What is the contact force between the boxes (the normal force that the two boxes exert on each other)?

Learn how to determine the contact force between a 1-kg and a 3-kg box when a 10-N force is applied. This problem explains Newton's second and third laws of motion, and provides a step-by-step solution for calculating acceleration and contact force. Suitable for high school physics students.

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