To determine the velocity of the package when it reaches the bottom of the chat, we can use the principles of conservation of energy. The package will undergo a loss in potential energy and a gain in kinetic energy as it moves from point A to the bottom of the chat.
1. Calculate the potential energy at point A:
Potential Energy (PE) = mass * gravity * height
PE = 6 kg * 9.8 m/s^2 * 0 m (since the height at point A is 0)
PE = 0 Joules
2. Calculate the kinetic energy at the bottom of the chat:
Kinetic Energy (KE) = (1/2) * mass * velocity^2
Since the package is released from rest, the initial velocity (v₀) is 0.
Therefore, KE = (1/2) * 6 kg * v^2
3. Apply the conservation of energy principle:
The loss in potential energy is equal to the gain in kinetic energy.
PE = KE
0 Joules = (1/2) * 6 kg * v^2
4. Solve for velocity (v):
0 = (1/2) * 6 kg * v^2
0 = 3 kg * v^2
v^2 = 0 m^2/s^2
v = 0 m/s
Therefore, the velocity of the package when it reaches the bottom of the chat is 0 m/s (since it already starts from rest).
Next, to determine the distance D from the bottom of the chat to point C where the package stops, we can calculate the net force acting on the package and use it to find the distance over which the package decelerates until it stops.
1. Calculate the net force:
Net Force (F_net) = Frictional Force (F_friction) - Weight (F_weight)
Frictional Force (F_friction) = coefficient of friction * Normal Force
Normal Force = mass * gravity
F_friction = 0.15 * (6 kg * 9.8 m/s^2)
F_friction = 8.82 N
Weight (F_weight) = mass * gravity
F_weight = 6 kg * 9.8 m/s^2
F_weight = 58.8 N
F_net = F_friction - F_weight
F_net = 8.82 N - 58.8 N
F_net = -49.98 N (negative sign indicates opposing direction)
2. Calculate the deceleration (a):
F_net = mass * acceleration
-49.98 N = 6 kg * a
a = -8.33 m/s^2 (again, negative sign indicates deceleration)
3. Calculate the distance (D):
Use the equation of motion: v^2 = v₀^2 + 2 * a * D
Since the package starts from rest at C, the final velocity (v) is 0.
Therefore, 0 = 0^2 + 2 * (-8.33 m/s^2) * D
Solving for D:
0 = -16.66 * D
D = 0 m
Therefore, the distance D from the bottom of the chat to point C where the package stops is 0 meters.