91Ó°ÊÓ

In physics and engineering, converting units is a fundamental skill. When dealing with springs, we often encounter a spring constant, denoted as \( k \), which describes how stiff the spring is. The spring constant is measured in units of force per length, like \( \,\text{Newtons per meter (N/m) or Newtons per centimeter (N/cm)} \).
Converting from one unit to another requires recognizing the relationship between meters and centimeters, where \( 1\, \text{cm} = 0.01 \text{m} \). Thus, to convert a spring constant given as \( 40.0\, \text{N/cm} \) to \( \text{N/m} \), you multiply by 100. This conversion reflects the fact that a meter is 100 times larger than a centimeter.
Consequently, the spring constant becomes \( 40.0 \times 100 = 4000 \, \text{N/m} \). This precise conversion ensures that all calculations involving distance and force are coherent in their units, critical for accurately computing energy values in subsequent steps.

Potential energy stored in a spring is a powerful concept in physics, especially when dealing with energy transformations. When a spring is compressed or stretched from its natural length, it holds potential energy, expressed as:

\[ PE = \frac{1}{2} k x^2 \]
Here:

• \( k \) is the spring constant in \( \text{N/m} \).
• \( x \) is the deviation from the spring's natural length, in meters.

The potential energy equation stems from Hooke's law, describing the effort needed to either compress or extend a spring.
In our exercise, compressing the spring by \( 0.375\, \text{m} \) with \( k = 4000 \text{N/m} \) results in \( PE = 281.25 \text{J} \). This energy highlights the amount of work done on compressing the spring, ready to be converted into kinetic energy or be partially retained as potential energy under different compression states.

Kinetic energy is the energy associated with motion. When stored potential energy from a spring is transformed, it becomes the kinetic energy of the moving sled and rider. The kinetic energy \( KE \) is calculated using the formula:

\[ KE = \frac{1}{2} mv^2 \]
Here:

• \( m \) is the mass of the sled and rider (70 kg).
• \( v \) is the velocity of the sled.

In the problem, when the spring releases its full potential energy, it completely converts into kinetic energy, allowing us to solve for the sled's velocity.
Using the conservation of energy principle:

• For a fully released spring, \( v_a = 2.83\, \text{m/s} \), as \( 281.25 \text{J} \) transforms entirely into kinetic energy.
• When the spring is still partially compressed (0.200 m), not all potential energy is converted. The leftover potential energy is subtracted from the total stored energy to compute \( v_b = 2.40 \text{m/s} \).

This approach demonstrates how energy conservation allows us to relate potential and kinetic energies effectively, using given mass and energy measurements.