A 20-kg block, initially moving 12 m/s, slides 5 meters across a rough horizontal surface from point A to point B, producing 400 J of thermal energy. It then slides along a frictionless surface from point B up the ramp to point C.
Find the speed of the block at the top of the ramp (point C). Show your work.
To find the speed of the block at the top of the ramp (point C), we can use the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant if no external forces act on it.
Let's break down the problem into different stages and calculate the total mechanical energy at each stage:
Stage 1: Sliding across the rough surface from point A to point B.
In this stage, the block is subjected to the force of friction, which results in the generation of thermal energy. This energy is not conserved and is lost from the system. The work done by the friction force is given by:
Work_done = force × distance = friction_force × distance
We are given that the distance is 5 meters and that 400 J of thermal energy is produced.
So, 400 J = friction_force × 5 m
Since work done is equal to the change in mechanical energy, we can express this equation as:
Work_done = change_in_mechanical_energy = final_mechanical_energy - initial_mechanical_energy
Since the block starts from rest at point A, the initial mechanical energy is zero. Therefore:
400 J = final_mechanical_energy - 0 J
final_mechanical_energy = 400 J
Stage 2: Sliding up the frictionless ramp from point B to point C.
In this stage, there is no friction, so no additional thermal energy is generated. Therefore the mechanical energy at point B will be equal to the mechanical energy at point C.
The mechanical energy is given by the sum of kinetic energy (KE) and potential energy (PE):
Mechanical_energy = KE + PE
At point B, the block has only kinetic energy. The kinetic energy is given by:
KE = 1/2 × mass × velocity^2
We are given that the mass of the block is 20 kg, and the initial velocity at point A is 12 m/s. Therefore:
KE_B = 1/2 × 20 kg × (12 m/s)^2
KE_B = 1/2 × 20 kg × 144 m^2/s^2
KE_B = 20 kg × 72 m^2/s^2
KE_B = 1440 J
Since mechanical energy is conserved, the mechanical energy at point C will also be 1440 J.
At point C, the block has both kinetic and potential energy. The potential energy is given by:
PE = mass × gravity × height
We are not given the height, but since the surface is frictionless, we can assume that the mechanical energy is conserved and therefore the height from point B to point C is the same as the vertical distance from point A to point B.
Substituting the values into the equation:
1440 J = mass × gravity × height
Plugging in the known values:
1440 J = 20 kg × 9.8 m/s^2 × height
1440 J = 196 kg·m^2/s^2 × height
Now, let's solve for the height:
height = 1440 J / (196 kg·m^2/s^2)
height = 7.346 m
The height from point B to point C is 7.346 meters.
Finally, we can calculate the speed (velocity) at point C by using the principle of conservation of mechanical energy:
Mechanical_energy_C = KE_C + PE_C
Since the mechanical energy is conserved, we know that:
Mechanical_energy_C = Mechanical_energy_B
KE_C + PE_C = KE_B + PE_B
We already calculated KE_B as 1440 J. And since there is no potential energy at point B, PE_B is zero.
KE_C + PE_C = 1440 J + 0 J
KE_C + PE_C = 1440 J
At point C, the block has only kinetic energy, so the potential energy will be zero. Therefore:
KE_C + 0 J = 1440 J
KE_C = 1440 J
Now, let's calculate the kinetic energy at point C, which is also the final mechanical energy:
KE_C = 1/2 × mass × velocity_C^2
We know the mass is 20 kg. Plugging in the known values:
KE_C = 1/2 × 20 kg × velocity_C^2
1440 J = 10 kg × velocity_C^2
Dividing both sides of the equation by 10 kg:
144 J/kg = velocity_C^2
Taking the square root of both sides:
velocity_C ≈ √144 J/kg
velocity_C ≈ 12 m/s
Therefore, the speed of the block at the top of the ramp (point C) is approximately 12 m/s.