In a circus performance, a monkey on a sled

is given an initial speed of 3.9 m/s up a 17

incline. The combined mass of the monkey
and the sled is 18.1 kg, and the coefficient
of kinetic friction between the sled and the
incline is 0.34.
The acceleration of gravity is 9.81 m/s
2
.
How far up the incline does the sled move?
Answer in units of m

To find the distance the sled moves up the incline, we can use the concept of work-energy theorem. The work done on the sled is equal to the change in its kinetic energy.

1. Calculate the net force acting on the sled:
The force pushing the sled up the incline is the component of the force due to gravity parallel to the incline. This force can be calculated using the following equation:
F_parallel = m * g * sin(θ)
Where:
m = mass of the monkey and sled (18.1 kg in this case)
g = acceleration due to gravity (9.81 m/s^2)
θ = angle of the incline (17 degrees)

2. Calculate the frictional force:
The frictional force can be calculated using the following equation:
F_friction = μ * m * g * cos(θ)
Where:
μ = coefficient of kinetic friction (0.34 in this case)
m = mass of the monkey and sled (18.1 kg)
g = acceleration due to gravity (9.81 m/s^2)
θ = angle of the incline (17 degrees)

3. Calculate the net force:
The net force can be calculated as the difference between the force pushing the sled up the incline and the frictional force:
Net force = F_parallel - F_friction

4. Calculate the work done on the sled:
The work done on the sled is equal to the net force multiplied by the distance traveled:
Work = Net force * Distance

5. Calculate the change in kinetic energy:
The change in kinetic energy can be calculated using the following equation:
ΔKE = (1/2) * m * v^2
Where:
m = mass of the monkey and sled (18.1 kg)
v = final velocity of the sled (zero in this case)

6. Equate the work done on the sled to the change in kinetic energy and solve for distance (d):
Net force * Distance = ΔKE

Now, plug in the known values and solve for distance (d).

Use conservation of energy

(Increase in potential energy) = (Initial kinetic energy) + (work done against friction)

Let L be the length it goes up the incline.
M*g*L*sin 17 = (1/2)*MVo^2 + M*g*L*cos17*(mu_k)

mu_k is the kinetic friction coefficient, 0.34

Cancel out the M's and solve for L