A 0.45kg shuffleboard puck is given an initial velocity down the playing surface of 4.5 m/s. If the coefficient of friction between the puck and the surface is 0.20, how far will the puck slide before coming to rest?
It will slide a distance X until the work done against friction equals the initial kinetic energy.
M*g*mu*X = (1/2) M V^2
X = V^2/(2*g*mu)
mu is the coefficient of kinetic friction, 0.20 in this case.
To determine how far the puck will slide before coming to rest, we need to calculate the frictional force acting on the puck and use it to determine the distance.
Step 1: Calculate the frictional force (F_friction):
The frictional force is given by the equation F_friction = coefficient of friction * normal force. The normal force can be calculated as normal force = mass * gravity, where the mass is 0.45 kg and gravity is 9.8 m/s^2.
normal force = 0.45 kg * 9.8 m/s^2 = 4.41 N
F_friction = 0.20 * 4.41 N = 0.882 N
Step 2: Calculate the deceleration (a):
The deceleration can be calculated using Newton's second law, F = m * a, where F is the frictional force and m is the mass of the puck.
0.882 N = 0.45 kg * a
a = 0.882 N / 0.45 kg = 1.96 m/s^2
Step 3: Calculate the distance (d):
The distance can be calculated using the kinematic equation, v_f^2 = v_i^2 + 2 * a * d, where v_f is the final velocity (0 m/s), v_i is the initial velocity (4.5 m/s), a is the acceleration (1.96 m/s^2), and d is the distance.
0^2 = (4.5 m/s)^2 + 2 * 1.96 m/s^2 * d
0 = 20.25 m^2/s^2 + 3.92 m/s^2 * d
3.92 m/s^2 * d = -20.25 m^2/s^2
d = (-20.25 m^2/s^2) / (3.92 m/s^2) = -5.17 m
Since distance cannot be negative, the puck will slide for approximately 5.17 meters before coming to rest.
To find the distance the puck will slide before coming to rest, we can use the concept of work-energy principle.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
The initial kinetic energy of the puck can be found using the formula:
KE_initial = (1/2) * mass * velocity^2
Plugging in the given values:
KE_initial = (1/2) * 0.45 kg * (4.5 m/s)^2
Next, we need to calculate the work done by friction. The work done by friction is given by the formula:
Work_done = force_of_friction * distance
The force of friction can be calculated using the formula:
force_of_friction = coefficient_of_friction * normal_force
The normal force is the force exerted by the surface on the puck and is equal to the weight of the puck. Thus, the normal force can be calculated as:
normal_force = mass * gravitational_acceleration
Substituting the given values:
normal_force = 0.45 kg * 9.8 m/s^2
Once we have the force of friction, we can calculate the work done by friction using:
Work_done = force_of_friction * distance
Since the puck comes to rest, the work done by friction is equal to the initial kinetic energy:
Work_done = KE_initial
Plugging in the values, we can solve for the distance:
distance = Work_done / force_of_friction
Now, we have all the information needed to calculate the distance the puck will slide before coming to rest. Let's calculate.
KE_initial = (1/2) * 0.45 kg * (4.5 m/s)^2
KE_initial = 4.5375 J
normal_force = 0.45 kg * 9.8 m/s^2
normal_force = 4.41 N
force_of_friction = 0.20 * 4.41 N
force_of_friction = 0.882 N
distance = 4.5375 J / 0.882 N
distance ≈ 5.14 m
Therefore, the puck will slide approximately 5.14 meters before coming to rest.