91Ó°ÊÓ

Spring potential energy is a type of energy stored in a compressed or stretched spring. It can be calculated using the formula:
\[ U = \frac{1}{2} k x^2 \]where:

• \( U \) is the spring potential energy (in Joules),
• \( k \) is the spring constant (in \( \mathrm{N/m} \)), a measure of the stiffness of the spring,
• \( x \) is the displacement from the spring's equilibrium position (in meters).

Let's consider the given problem: the spring initially compresses by 0.375 meters. By substituting \( k = 4000 \, \mathrm{N/m} \) and \( x = 0.375 \, \mathrm{m} \), we find that the initial spring potential energy is a significant value that will eventually help propel the sled forward. It's crucial to understand that once the spring releases, this potential energy transforms into kinetic energy. This is the essence of the energy conservation we're using here, ensuring that no energy is lost in a frictionless scenario.

Kinetic energy represents the energy that an object possesses due to its motion. The formula to calculate kinetic energy is:
\[ K = \frac{1}{2} mv^2 \]where:

• \( K \) is the kinetic energy (in Joules),
• \( m \) is the mass of the object (in kilograms),
• \( v \) is the velocity of the object (in meters per second).

In this particular exercise, the sled starts from rest, meaning its initial kinetic energy is zero. As the spring returns to its uncompressed state, the potential energy stored in the spring is converted entirely into kinetic energy. Hence, the sled gains speed. For part (a), we observe that the energy conversion is complete at the full release of the spring, resulting in the sled's highest speed for this configuration. Part (b) shows that when the spring is still partially compressed, some potential energy remains, leading to a lower velocity of the sled compared to complete decompression.

In the context of this problem, the sled moves on a frictionless surface. This assumption simplifies calculations because it eliminates energy losses due to friction—a force that opposes motion and dissipates energy as heat. On a frictionless surface:

• The total mechanical energy remains constant, assuming no external work is done.
• All potential energy from the spring is transformed into kinetic energy without losses.
• It allows us to apply the principle of conservation of energy directly without considering non-conservative forces.

The frictionless surface thus plays a key role by enabling the straightforward conversion of spring potential energy into kinetic energy. This ensures that the sled gains maximum possible speed as it is propelled across the surface. It serves an ideal scenario to apply theoretical principles of physics such as energy conservation effectively. This makes solving such exercises more manageable and provides a clear demonstration of these fundamental physical laws.