Question

At a waterpark, sleds with riders are sent along a slippery, horizontal surface by the release of a large compressed spring. The spring, with force constant \(k\) = 40.0 N/cm and negligible mass, rests on the frictionless horizontal surface. One end is in contact with a stationary wall. A sled and rider with total mass 70.0 kg are pushed against the other end, compressing the spring 0.375 m. The sled is then released with zero initial velocity. What is the sled's speed when the spring (a) returns to its uncompressed length and (b) is still compressed 0.200 m?

Short answer

(a) 2.834 m/s, (b) 2.398 m/s.

Step-by-step solution

Convert Spring Constant

The force constant is given in N/cm, so we need to convert it to N/m. Since there are 100 cm in a meter, multiply the given k by 100:\[ k = 40.0 \, \text{N/cm} \times 100 \, \text{cm/m} = 4000 \, \text{N/m} \]

Calculate Initial Potential Energy

Use the formula for the potential energy stored in a compressed spring: \(E_{p1} = \frac{1}{2} k x^2\). Substitute in the converted spring constant and the compression distance:\[ E_{p1} = \frac{1}{2} \times 4000 \, \text{N/m} \times (0.375 \, \text{m})^2 = 281.25 \, \text{J} \]

Apply Conservation of Energy (Uncompressed)

When the spring returns to its uncompressed length, all potential energy is converted to kinetic energy. Use the formula \(E_k = \frac{1}{2} mv^2\) and set it equal to the potential energy calculated.\[ \frac{1}{2} \times 70.0 \, \text{kg} \times v^2 = 281.25 \, \text{J} \]\[ v^2 = \frac{281.25 \, \text{J}}{35.0 \, \text{kg}} = 8.036 \]\[ v = \sqrt{8.036} \approx 2.834 \, \text{m/s} \]

Calculate Potential Energy (Compressed)

When the spring is still compressed by 0.200 m, calculate the remaining potential energy in the spring.\[ E_{p2} = \frac{1}{2} \times 4000 \, \text{N/m} \times (0.200 \, \text{m})^2 = 80.0 \, \text{J} \]

Apply Conservation of Energy (Compressed)

Apply conservation of energy to find the kinetic energy when the spring is compressed by 0.200 m. The initial potential energy minus the remaining potential energy equals the kinetic energy.\[ E_k = E_{p1} - E_{p2} = 281.25 \, \text{J} - 80.0 \, \text{J} = 201.25 \, \text{J} \]

Calculate Speed (Compressed)

Convert the kinetic energy (when the spring is compressed by 0.200 m) into speed using \(E_k = \frac{1}{2} mv^2\).\[ \frac{1}{2} \times 70.0 \, \text{kg} \times v^2 = 201.25 \]\[ v^2 = \frac{201.25}{35.0} = 5.75 \]\[ v = \sqrt{5.75} \approx 2.398 \, \text{m/s} \]

Key concepts

Potential Energy

Potential energy is the energy stored within an object due to its position or configuration. For springs, potential energy is stored during compression or extension. This is described by Hooke's Law, which states that the force exerted by a spring is proportional to its displacement. In mathematical terms, the potential energy (\(E_p\)) stored in a spring is given by the formula:
\[ E_p = \frac{1}{2} k x^2 \]where:

• \(k\)is the spring constant, indicating the spring's stiffness.
• \(x\)is the displacement from the spring's natural length.

In our exercise, when the spring is compressed by 0.375 meters, we store potential energy, calculated using the spring constant and the displacement. This energy is what allows the sled to accelerate once released.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It's a vital concept when considering how energy transforms. As potential energy stored in a spring converts to kinetic energy, you see the principles of energy conservation in action.
The formula for kinetic energy (\(E_k\)) is:
\[ E_k = \frac{1}{2} mv^2 \]where:

• \(m\)is the mass of the object.
• \(v\)is its velocity.

In our scenario, when the spring is released, the stored potential energy converts into kinetic energy, setting the sled into motion with a velocity calculated from the energy equations. This demonstrates how energy can be moved from potential to kinetic as the system evolves.

Spring Constant

The spring constant, denoted as (\(k\)), measures a spring's stiffness, dictating how much force is needed for a certain displacement. A stiffer spring has a higher spring constant and thus stores more energy for the same compression or extension.
Springs with a high spring constant return more force when compressed, leading to quicker acceleration of attached objects. This is directly visible when converting potential energy into kinetic energy.
In our exercise, the spring constant was initially in N/cm, and converting it to N/m is crucial for calculations, ensuring consistent units across measurements. With a spring constant of 4000 N/m after conversion, it highlights the energetic capacity and influences both potential and kinetic energies as the spring operates.