A horizontal spring with spring constant 250 N/m is compressed by 12 cm and then used to launch a 250 g box across the floor. The coefficient of kinetic friction between the box and the floor is 0.23. What is the box's launch speed?

Verified step by step guidance

1. Step 1: Calculate the elastic potential energy stored in the compressed spring using the formula \( U = \frac{1}{2} k x^2 \), where \( k \) is the spring constant (250 N/m) and \( x \) is the compression distance (12 cm, converted to meters as 0.12 m).
2. Step 2: Determine the work done by friction as the box moves across the floor using the formula \( W_{friction} = f_k \cdot d \), where \( f_k = \mu_k \cdot m \cdot g \) is the kinetic friction force. Here, \( \mu_k \) is the coefficient of kinetic friction (0.23), \( m \) is the mass of the box (250 g, converted to kg as 0.25 kg), and \( g \) is the acceleration due to gravity (9.8 m/s^2).
3. Step 3: Apply the work-energy principle. The initial elastic potential energy stored in the spring is converted into the kinetic energy of the box and the work done against friction. Use the equation \( U - W_{friction} = \frac{1}{2} m v^2 \), where \( v \) is the launch speed of the box.
4. Step 4: Rearrange the equation from Step 3 to solve for \( v \), the launch speed. The formula becomes \( v = \sqrt{\frac{2 (U - W_{friction})}{m}} \).
5. Step 5: Substitute the values for \( U \), \( W_{friction} \), and \( m \) into the equation from Step 4 to calculate the box's launch speed. Ensure all units are consistent (meters, kilograms, seconds).

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 250 N/m, and the spring is compressed by 0.12 m, allowing us to calculate the potential energy stored in the spring.

Kinetic Energy and Work-Energy Principle

The kinetic energy (KE) of an object is given by the formula KE = 0.5mv², where m is the mass and v is the velocity. 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 potential energy from the spring is converted into kinetic energy of the box, minus the work done against friction.

Friction and Normal Force

Friction is a force that opposes the motion of an object and is calculated as F_friction = μN, where μ is the coefficient of friction and N is the normal force. For the box on the floor, the normal force equals its weight (mg). The coefficient of kinetic friction (0.23) will determine how much energy is lost to friction as the box moves, affecting its final launch speed.