A 4kg box starts up a 30 degree incline with 128J of kinetic energy. How far will it slide up the plane if the co-efficient of friction is 0.3?

To determine how far the box will slide up the incline, we can start by analyzing the forces acting on the box and applying the principle of conservation of energy.

1. First, let's analyze the forces acting on the box:
- The weight of the box acts straight downward and can be calculated as W = m * g, where m is the mass of the box (4 kg) and g is the acceleration due to gravity (9.8 m/s^2).
- The normal force, N, acts perpendicular to the plane and counterbalances the component of the weight that is perpendicular to the plane. It can be calculated as N = m * g * cos(theta), where theta is the angle of the incline (30 degrees).
- The force of friction, F_friction, acts parallel to the incline and opposes the motion of the box. It can be calculated as F_friction = coefficient_of_friction * N, where the coefficient_of_friction is given as 0.3.

2. Now, let's calculate the work done by the friction force:
- The work done, W_friction, can be calculated as W_friction = F_friction * d, where d is the displacement of the box up the incline.

3. Using the principle of conservation of energy:
- The initial kinetic energy of the box is given as 128J.
- The final potential energy of the box when it reaches the maximum height is given as m * g * h, where h is the vertical height above the starting point.

4. Set up an equation equating the initial kinetic energy to the final potential energy minus the work done by friction:
(1/2) * m * v^2 = m * g * h - W_friction

5. Rearrange the equation and substitute the given values:
(1/2) * m * v^2 + W_friction = m * g * h
(1/2) * 4 * v^2 + 0.3 * m * g * cos(theta) * d = 4 * g * h

6. Now, solve for d:
Substitute the given values: m = 4 kg, coefficient_of_friction = 0.3, g = 9.8 m/s^2, and theta = 30 degrees.
Plug these values into the equation and solve for d.

Using these steps, you can determine the distance the box will slide up the incline.