In a circus performance, a monkey on a sled is given an initial speed of 4.0 m/s up a 32° incline. The combined mass of the monkey and the sled is 19.1 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?
To find how far up the incline the sled moves, we first need to determine the net force acting on the sled. Then we can use this information to calculate the distance traveled up the incline.
Step 1: Calculate the gravitational force acting on the sled.
The gravitational force is calculated using the formula:
F_grav = m * g
Where:
m is the combined mass of the monkey and the sled (19.1 kg)
g is the acceleration due to gravity (9.8 m/s^2)
Substituting the given values:
F_grav = 19.1 kg * 9.8 m/s^2 = 187.38 N
Step 2: Calculate the force of friction acting on the sled.
The force of friction can be calculated using the formula:
F_friction = μ * F_normal
Where:
μ is the coefficient of kinetic friction (0.20)
F_normal is the normal force acting on the sled, which is equal to the component of the gravitational force perpendicular to the incline.
F_normal = m * g * cos(θ)
Where:
θ is the angle of the incline (32°)
Substituting the given values:
F_normal = 19.1 kg * 9.8 m/s^2 * cos(32°) ≈ 161.25 N
Now, we can calculate the force of friction:
F_friction = 0.20 * 161.25 N = 32.25 N
Step 3: Calculate the net force along the incline.
The net force along the incline is calculated as:
F_net = F_applied - F_friction
Since no additional force is being applied to the sled, the applied force is zero:
F_applied = 0 N
F_net = 0 N - 32.25 N = -32.25 N (negative because it opposes the motion)
Step 4: Calculate the acceleration of the sled.
The acceleration of the sled is given by Newton's second law:
F_net = m * a
Substituting the given values:
-32.25 N = 19.1 kg * a
Solving for a:
a = -1.686 m/s^2 (negative because it is opposing the motion)
Step 5: Calculate the distance traveled up the incline.
The distance traveled can be found using the following equation of motion:
d = (v_f^2 - v_i^2) / (2 * a)
Where:
v_f is the final velocity (0 m/s, as the sled comes to rest)
v_i is the initial velocity (4.0 m/s)
a is the acceleration (-1.686 m/s^2)
Substituting the given values:
d = (0^2 - 4.0^2) / (2 * (-1.686)) = 2.38 m
Therefore, the sled moves approximately 2.38 meters up the incline