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To solve this problem, let's analyze the setup and work through the steps.

### Problem Setup
- A block of mass \( m \) is placed on a frictionless horizontal surface.
- It is connected to two identical springs of spring constant \( k \) on each side.
- One end of the left spring is attached to a wall, while the right spring is being pulled with constant velocity \( u \).
- Initially, the springs are at their natural (unstretched) lengths.

We need to find the minimum time \( t \) after which the block attains a speed equal to \( u \).

### Solution Approach

1. **Relative Motion with Springs:**
   Since the right spring is being pulled with velocity \( u \), the effective compression or extension on the block from the right spring will be determined by this pulling velocity. The block will begin to accelerate as a result of the restoring forces from the springs.

2. **Spring Force and Acceleration:**
   As the right end is pulled with velocity \( u \), the right spring will stretch, while the left spring will experience a compressive force. The net restoring force \( F \) on the block will be due to the combined force exerted by both springs.

3. **Equation of Motion:**
   The force \( F \) due to the spring is given by Hooke’s Law:
   \[
   F = -2kx
   \]
   where \( x \) is the displacement of the block from the equilibrium position (which initially is zero).

4. **Condition for the Block's Velocity:**
   The block needs to achieve a velocity \( u \). As the end of the right spring moves with velocity \( u \), the block’s velocity will increase from zero until it matches this speed.

5. **Energy Conservation Approach:**
   Since the system is frictionless, we can equate the kinetic energy of the block to the work done by the spring forces.

6. **Result:**
   After analysis, it can be shown that the minimum time \( t \) required for the block to reach speed \( u \) is:
   \[
   t = \frac{m}{2k}
   \]

### Final Answer
\[
t = \frac{m}{2k}
\]

Would you like a detailed explanation of each step, or do you have any questions?

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Here are some related questions for further understanding:

1. What happens to the block's acceleration as the right spring continues to stretch?
2. How would the solution change if the surface was not frictionless?
3. What is the role of the spring constant \( k \) in the time calculation?
4. If the block’s mass \( m \) doubled, how would the minimum time change?
5. How does the equilibrium position of the block shift as the right spring is stretched?

**Tip:** In such spring-mass systems, energy conservation is often a powerful tool to analyze motion and forces.

A block of mass m placed on a frictionless horizontal floor is connected with two identical springs each of force constant k. One end of the left spring is connected to a fixed support and one end of the right spring is free. Initially the block is at rest, the springs are collinear and relaxed. If someone begins to pull the free end of the right spring with a constant velocity u away from the wall, the minimum time after which the block acquires a speed equal to u?

Understand how to calculate the minimum time for a block to reach a specific speed in a spring-mass system on a frictionless floor. Explore concepts such as Hooke's Law, oscillations, and energy conservation. This problem involves pulling a spring with a constant velocity to determine the time required for the block to acquire the same speed.

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