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When a spring is compressed or extended, it stores energy. This energy is known as elastic potential energy. The work done by a spring is essentially the transfer of this potential energy. The formula to calculate the work done by a spring is: \[ W = \frac{1}{2} k x^2 \]where:- \( W \) is the work done by the spring,- \( k \) is the spring constant (a measure of the spring's stiffness),- \( x \) is the distance the spring is compressed or stretched from its natural length.In practice, this means that if you compress or stretch a spring, you're investing energy into it. When you release the spring, this energy is transferred as work to another object, like the block of ice in our example. The spring's job is done when it returns to its original shape, and it has transferred all its stored energy.

Potential energy can be thought of as stored energy ready to be converted into work. In our scenario, when the spring is compressed, it stores energy in the form of elastic potential energy.The potential energy in a compressed spring can be calculated with the same equation we use for work:\[ PE_{spring} = \frac{1}{2} k x^2 \]Thus, potential energy depends on both the spring constant and the extent of compression or expansion.

• If the spring constant \( k \) is larger, the spring is stiffer, and it can store more energy.
• If the compression \( x \) is greater, the potential energy stored is also greater.

This energy can be released to perform work on objects, as seen in the acceleration of the ice block after the spring is released. It transitions into kinetic energy, propelling the mass forward.

Kinetic energy refers to the energy an object has due to its motion. When the spring in our problem is released, the potential energy stored in the spring is transformed into kinetic energy.The formula for kinetic energy is:\[ KE = \frac{1}{2} m v^2 \]where:- \( KE \) is the kinetic energy,- \( m \) is the mass of the object (like our 4 kg block of ice),- \( v \) is the velocity of the object.After the spring has done its work, it transfers its potential energy to the block, converting it into kinetic energy and giving the block speed. By equating the spring's potential energy with the block's kinetic energy, you can solve for the block's speed after being pushed by the spring.

Performing physics calculations involves translating real-world scenarios into mathematical equations to predict outcomes. Here's how it is typically done:1. **Identify the Type of Energy in Play**: In the spring example: - Start with potential energy when the spring is compressed.2. **Apply the Relevant Formulas**: - Use the potential energy formula \( PE = \frac{1}{2} k x^2 \). - Calculate how this energy will transform when released.3. **Energy Conversion and Conservation**: - Utilize the concept of energy conversion. Potential energy is turned into kinetic energy. - Use the kinetic energy formula \( KE = \frac{1}{2} m v^2 \) for further calculations.4. **Solve Step-by-Step**: - Substitute values for constants \( k \) and \( m \), and calculate accordingly. - Perform algebraic manipulations to find unknowns like velocity.These calculations help build an understanding of how forces and energy interact in the physical world, providing clear predictions of an object's motion and speed.