A horizontal spring with spring constant 110N/m is compressed 16cm and used to launch a 2.8kg 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.

try kx=Ffmgh , then solve for h , (its an idea)

To find how far the box slides across the rough surface before stopping, we need to first determine the initial energy stored in the compressed spring and then calculate the work done against friction.

1. Calculate the initial potential energy stored in the compressed spring:
- The potential energy stored in a spring is given by the equation: PE = (1/2) k x^2, where PE is the potential energy, k is the spring constant, and x is the compression distance.
- In this case, the spring constant (k) is 110 N/m and the compression distance (x) is 16 cm. Convert the distance to meters by dividing by 100: x = 16 cm / 100 = 0.16 m.
- Substitute the values into the formula:
PE = (1/2) * 110 N/m * (0.16 m)^2
PE = 0.88 J

2. Calculate the work done against friction:
- The work done against friction can be determined using the formula: work = force * distance.
- The force of friction (Ff) is given by the equation: Ff = coefficient of kinetic friction * normal force. The normal force (Fn) is equal to the weight of the box, which can be calculated as Fn = mass * gravity, where the mass is 2.8 kg and gravity is 9.8 m/s^2.
Fn = 2.8 kg * 9.8 m/s^2
Fn = 27.44 N
- Substitute the values into the equation:
Ff = 0.15 * 27.44 N
Ff = 4.116 N
- Now, to calculate the work done against friction, we need to find the distance over which the friction acts. Let's call this distance S (the distance the box slides).
- The work done against friction (Wf) can be determined by: Wf = Ff * S.

3. Equate the initial potential energy to the work done against friction:
- Since work done is the change in energy, we have:
PE = Wf
0.88 J = Ff * S
S = 0.88 J / 4.116 N
S ≈ 0.213 m

Therefore, the box slides approximately 0.213 meters (or 21.3 cm) across the rough surface before coming to a stop.