A horizontal spring with spring constant 100 N/m is compressed 20 cm and used to launch a 2.5 kg box across a frictionless, horizontal surface. After the box travels some distance, the surface becomes rough. The coefficient of kinetic friction of the box on the surface is 0.15. Use work and energy to find how far the box slides across the rough surface before stopping.

Verified step by step guidance

1. Step 1: Calculate the initial elastic potential energy stored in the spring using the formula \( U_s = \frac{1}{2} k x^2 \), where \( k \) is the spring constant (100 N/m) and \( x \) is the compression of the spring (0.20 m).
2. Step 2: Recognize that the initial elastic potential energy of the spring is converted into the kinetic energy of the box as it leaves the spring, and then into work done against friction as the box slides on the rough surface. The work-energy principle states \( U_s = W_f \), where \( W_f \) is the work done by friction.
3. Step 3: Express the work done by friction as \( W_f = f_k d \), where \( f_k \) is the kinetic friction force and \( d \) is the distance the box slides. The kinetic friction force is given by \( f_k = \mu_k m g \), where \( \mu_k \) is the coefficient of kinetic friction (0.15), \( m \) is the mass of the box (2.5 kg), and \( g \) is the acceleration due to gravity (9.8 m/s^2).
4. Step 4: Substitute \( f_k \) into the work equation \( W_f = f_k d \), and set it equal to the initial elastic potential energy \( U_s \). Solve for \( d \), the distance the box slides: \( d = \frac{U_s}{f_k} \).
5. Step 5: Plug in the values for \( U_s \) (from Step 1) and \( f_k \) (from Step 3) into the equation for \( d \) to find the distance the box slides before stopping. Ensure all units are consistent during calculations.

Key Concepts

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to its displacement from the equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. In this scenario, the spring constant is 100 N/m, and the spring is compressed by 0.2 m, allowing us to calculate the potential energy stored in the spring, which is converted into kinetic energy when the box is launched.

Work-Energy Principle

The Work-Energy Principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the initial kinetic energy of the box, derived from the spring's potential energy, will be reduced by the work done against friction as the box slides across the rough surface. This principle allows us to relate the initial energy to the distance traveled before the box comes to a stop.

Friction and Kinetic Energy Loss

Friction is a force that opposes the motion of an object, and the coefficient of kinetic friction quantifies this resistance. The work done by friction can be calculated using the formula W_friction = f_friction * d, where f_friction is the frictional force and d is the distance traveled. In this problem, the kinetic energy lost due to friction will determine how far the box slides before stopping, allowing us to find the distance using the relationship between work and energy.