An 18 kg curling stone sliding on a sheet of ice hits a rough patch on the ice where the coefficient of friction increases from 0.13 to 0.28. It enters this rough patch with a speed of 2.2 m/s and leaves with

a speed of 1.4 m/s. How long is the rough patch?

What are we suppose to do in this question?

work done = force * distance = change in kinetic energy

F = -.28 m g = -.28 * 18 * 9.81
work = F * x

final - initial Ke

= (1/2) *18 * (.13^2 - .28^2)

so
-.28 * 9.81 = (1/2) * (.13^2 - .28^2)

forgot x

-.28 * 9.81 * x = (1/2) * (.13^2 - .28^2)

To solve this question, we can use the concept of Newton's second law of motion and the work-energy principle.

First, let's break down the steps to solve the problem:

1. Calculate the acceleration of the curling stone within the rough patch using the change in velocity and the time taken.
- Recall that acceleration (a) is the change in velocity (Δv) divided by the time (t): a = Δv / t.

2. Apply Newton's second law of motion to find the net force acting on the curling stone while it is within the rough patch.
- Newton's second law states that the net force (F_net) acting on an object is equal to the mass (m) of the object multiplied by its acceleration (a): F_net = m * a.

3. Determine the frictional force (F_friction) acting on the curling stone within the rough patch using the coefficient of friction and the normal force.
- The frictional force (F_friction) can be calculated as the product of the coefficient of kinetic friction (μ_kinetic) and the normal force (F_normal): F_friction = μ_kinetic * F_normal.
- The normal force (F_normal) is equal to the gravitational force acting on the curling stone, which can be calculated as the weight (mg) of the stone: F_normal = mg.

4. Use the work-energy principle to relate the frictional force and the distance traveled within the rough patch.
- The work done by the frictional force (W_friction) is equal to the product of the frictional force and the distance traveled (d): W_friction = F_friction * d.
- The work done by a force can also be expressed as the change in kinetic energy, which is the difference between the initial kinetic energy (KE_initial) and the final kinetic energy (KE_final): W_friction = KE_final - KE_initial.

5. Equate the expressions for the work done by the frictional force and solve for the distance traveled within the rough patch (d).
- Set the equations for the work done by friction and the change in kinetic energy equal to each other: F_friction * d = KE_final - KE_initial.
- Rearrange the equation to solve for the distance (d): d = (KE_final - KE_initial) / F_friction.

By following these steps and plugging in the given values, you can calculate the length of the rough patch.