As shown in the figure below, two masses m1 = 4.10 kg and m2 which has a mass 50.0% that of m1, are attached to a cord of negligible mass which passes over a frictionless pulley also of negligible mass. If m1 and m2 start from rest, after they have each traveled a distance h = 1.10 m, use energy content to determine the following.

Hmmm. Something is missing here.

When people do not even go back and see if what they typed makes any sense, I just skip to the next question.

To determine the following, we need to use the principle of conservation of mechanical energy.

1. Find the potential energy of mass m1 at the starting position:
Potential energy (U1) = mass (m1) * gravitational acceleration (g) * height (h)
U1 = 4.10 kg * 9.8 m/s^2 * 1.10 m

2. Find the kinetic energy of mass m1 at the end position:
Since mass m1 is at a lower height h, its potential energy is converted into kinetic energy.
Kinetic energy (K1) = (1/2) * mass (m1) * velocity (v1)^2

3. Find the potential energy of mass m2 at the starting position:
Since m2 is attached to m1 and they move together, they share the same height h.
Potential energy (U2) = mass (m2) * gravitational acceleration (g) * height (h)
Since m2 has a mass 50.0% of m1, m2 = 0.50 * m1
U2 = (0.50 * 4.10 kg) * 9.8 m/s^2 * 1.10 m

4. Find the kinetic energy of mass m2 at the end position:
Since mass m2 is at a higher height h, its potential energy increases.
Kinetic energy (K2) = (1/2) * mass (m2) * velocity (v2)^2

5. Use the conservation of mechanical energy principle:
The total mechanical energy at the starting position (E1) is equal to the total mechanical energy at the end position (E2).
E1 = U1 + U2
E2 = K1 + K2

Now we can calculate the values step by step.

To determine the following using energy considerations, we need to find the potential energy and kinetic energy of the system at two different positions: before and after they have traveled a distance of 1.10 m.

Step 1: Determine the potential energy before and after displacement:
The potential energy (PE) of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the vertical displacement. Since both m1 and m2 start from rest, the initial potential energy of the system is 0.

1. Before displacement:
The potential energy of m1 before displacement is PE_1_initial = m1 * g * 0 = 0.
The potential energy of m2 before displacement is PE_2_initial = m2 * g * 0 = 0.

2. After displacement:
The potential energy of m1 after displacement is PE_1_final = m1 * g * h.
The potential energy of m2 after displacement is PE_2_final = m2 * g * h.

Step 2: Determine the kinetic energy after displacement:
The kinetic energy (KE) of an object is given by the formula KE = (1/2) * m * v^2, where m is the mass and v is the velocity.

1. After displacement:
Both masses, m1 and m2, have traveled a distance h. Hence, their velocities will be the same. Let's call this velocity v_final for both masses.

The kinetic energy of m1 after displacement is KE_1_final = (1/2) * m1 * v_final^2.
The kinetic energy of m2 after displacement is KE_2_final = (1/2) * m2 * v_final^2.

Step 3: Equating the initial and final energy:
According to the Law of Conservation of Energy, the total mechanical energy (potential energy + kinetic energy) of a system remains constant as long as no external forces are acting.

1. Initially (before displacement):
The total mechanical energy is E_initial = PE_1_initial + PE_2_initial + KE_1_initial + KE_2_initial.
Since the masses are at rest initially, their initial kinetic energies (KE_1_initial and KE_2_initial) are both 0, so the total mechanical energy is E_initial = PE_1_initial + PE_2_initial.

2. Finally (after displacement):
The total mechanical energy is E_final = PE_1_final + PE_2_final + KE_1_final + KE_2_final.

According to the Law of Conservation of Energy, E_initial = E_final.

Step 4: Substitute the values and solve for the unknowns:
Since we are given the values for m1, h, and the ratio of m2 to m1, we can substitute these into the above equations to solve for the unknowns.

Given:
m1 = 4.10 kg,
m2 = 50.0% of m1 (which means m2 = 0.5 * m1 = 0.5 * 4.10 kg = 2.05 kg),
h = 1.10 m.

Substituting these values into the equations derived above, you can solve for the final potential energy, final kinetic energy, and the final velocity of each mass after they have traveled a distance of 1.10 m.