In a circus performance, a monkey on a sled is given an initial speed of 3.2 m/s up a 31° incline. The combined mass of the monkey and the sled is 20.7 kg, and the coefficient of kinetic friction between the sled and the incline is 0.20. How far up the incline does the sled move?

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To find how far up the incline the sled moves, we can use the principles of physics, specifically the laws of motion and the concept of work-kinetic energy theorem.

Here are the steps to solve the problem:

Step 1: Resolve the force acting parallel and perpendicular to the incline.

The weight of the sled and monkey can be resolved into two components: one perpendicular to the incline (mg*cosθ) and one parallel to the incline (mg*sinθ), where m is the mass and θ is the angle of the incline.

Perpendicular component: mg*cosθ = 20.7 kg * 9.8 m/s^2 * cos(31°)
Parallel component: mg*sinθ = 20.7 kg * 9.8 m/s^2 * sin(31°)

Step 2: Calculate the force of kinetic friction.

The force of kinetic friction can be calculated using the equation: friction = coefficient of friction * perpendicular component.

Friction = 0.20 * (20.7 kg * 9.8 m/s^2 * cos(31°))

Step 3: Calculate the net force.

The net force is the difference between the parallel component and the force of kinetic friction.

Net force = parallel component - friction

Step 4: Use the work-kinetic energy theorem.

The work done on an object is equal to the change in its kinetic energy. In this case, the work done against the force of friction will be equal to the change in kinetic energy and can be calculated using the equation: work = net force * distance.

We can rearrange this equation to find the distance traveled: distance = work / net force.

Step 5: Calculate the distance.

Substitute the values for net force and work into the equation to find the distance traveled.

Distance = work / net force

Now let's calculate the values:

Perpendicular component: mg * cosθ = 20.7 kg * 9.8 m/s^2 * cos(31°) ≈ 178.04 N
Parallel component: mg * sinθ = 20.7 kg * 9.8 m/s^2 * sin(31°) ≈ 98.11 N
Friction = 0.20 * (20.7 kg * 9.8 m/s^2 * cos(31°)) ≈ 35.61 N
Net force = parallel component - friction = 98.11 N - 35.61 N ≈ 62.50 N

Given that the sled has an initial speed of 3.2 m/s, we can calculate the work done against friction using the formula: work = (1/2) * m * v², where v is the initial speed.

Work = (1/2) * 20.7 kg * (3.2 m/s)² ≈ 105.41 J

Distance = work / net force = 105.41 J / 62.50 N ≈ 1.69 meters

Therefore, the sled moves approximately 1.69 meters up the incline.