Buddy is unloading 7kg cans using a 45 degree downward ramp. The ramp was made slippery by the a detergent spill before. Buddy can push the cans off of his truck with an initial velocity of .2 m/s and it moves as 1.2 m/a by the time it reaches the lowest point on the ramp. The next week the detergent has been cleaned up and buddy realizes that a can pushed at 0.2m/s will come to a halt just before the lowest point of the ramp. What is the magnitude of kinetic friction between the can and the clean ramp?

To find the magnitude of kinetic friction between the can and the clean ramp, we can use the concept of conservation of mechanical energy.

Let's start by analyzing the situation when the ramp was slippery. When Buddy pushed the cans off the truck with an initial velocity of 0.2 m/s, the can accelerated and moved at a constant acceleration of 1.2 m/s² before reaching the lowest point of the ramp.

The can loses gravitational potential energy as it moves down the ramp due to its height change, but gains an equal amount of kinetic energy. Since the ramp is frictionless (due to the detergent spill), no energy is lost due to friction.

The gravitational potential energy (PE) of the can at the top of the ramp can be calculated using the formula PE = mgh, where m is the mass of the can, g is the acceleration due to gravity, and h is the height of the ramp. In this case, the height of the ramp is not given in the question, so we'll need to determine it later.

The kinetic energy (KE) of the can at the lowest point of the ramp is given by KE = 0.5mv², where m is the mass of the can and v is its velocity.

Since energy is conserved, we can set the gravitational potential energy at the top equal to the kinetic energy at the bottom:

mgh = 0.5mv²

The mass of the can cancels out, so we are left with:

gh = 0.5v²

Now, let's consider the situation when the ramp is clean (no more detergent spill). Buddy notices that a can pushed at 0.2 m/s will come to a halt just before the lowest point of the ramp. This means that the work done by kinetic friction is equal to the change in kinetic energy.

The work done by kinetic friction can be calculated using the formula W = f * d * cosθ, where W is the work, f is the magnitude of the force of kinetic friction, d is the distance over which the force is applied, and θ is the angle between the force and the displacement.

In this case, since the can comes to a halt, the change in kinetic energy is zero. Therefore, the work done by kinetic friction is also zero.

W = f * d * cosθ = 0

Since cosθ will be positive for a downward slope, we have:

f * d = 0

This implies that either the force of friction (f) is zero or the distance (d) is zero. Since the distance cannot be zero (as Buddy pushes the can along the ramp), the force of friction must be zero when the can comes to a halt.

Since the force of friction is zero, the net force acting on the can (in the direction opposite to motion) is also zero. The only other force acting on the can in this direction is the gravitational force (mg).

Therefore, the magnitude of the kinetic friction between the can and the clean ramp is equal to the gravitational force acting on the can, which is mg.

In summary, the magnitude of the kinetic friction between the can and the clean ramp is equal to the gravitational force acting on the can, which is mg.