Two blocks with masses m1 = 1.20 kg and m2 = 2.90 kg are connected by a massless string, as shown in the Figure. They are released from rest. The coefficent of kinetic friction between the upper block and the surface is 0.300. Assume that the pulley has a negligible mass and is frictionless, and calculate the speed of the blocks after they have moved a distance 74.0 cm.

To determine the speed of the blocks after they have moved a certain distance, we can use the principle of conservation of mechanical energy. We can calculate the potential energy lost by the system as the blocks move down and convert it into kinetic energy.

First, find the initial potential energy of the system. The potential energy is given by the equation PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height. In this case, the initial potential energy is equal to the sum of the potential energies of each block.

PE_initial = m1gh1 + m2gh2

Next, find the final potential energy of the system. Since the blocks are released from rest, the potential energy is converted entirely into kinetic energy.

KE_final = KE1 + KE2

Finally, equate the initial potential energy to the final kinetic energy to determine the speed of the blocks.

PE_initial = KE_final

To find the height, h1 and h2, we need to consider the distance the blocks have moved. In this case, the total distance the blocks have moved is given as 74.0 cm. Since the string moves with the same velocity, we can write the following relationship:

h1 - h2 = distance

We know that the mass m1 = 1.20 kg and m2 = 2.90 kg.

Now, let's start solving the problem.

1. Calculate the height difference:
h1 - h2 = 74.0 cm

2. Convert the height difference to meters:
h1 - h2 = 74.0 cm = 0.74 m

3. Substitute the values for the masses, acceleration due to gravity, and height difference into the equation:
(1.20 kg) * g * h1 - (2.90 kg) * g * h2 = 0.74 m

4. Use the coefficient of kinetic friction to determine the frictional force acting on the top block:
Frictional force = coefficient of kinetic friction * Normal force

The normal force can be determined by multiplying the mass of the block by the acceleration due to gravity:
Normal force = m1 * g

5. Substitute the values for the mass of the block, coefficient of kinetic friction, and acceleration due to gravity into the equation:
Frictional force = (1.20 kg) * 0.3 * g

6. The net force on the block can be given as:
Net force = m1 * a1

7. Substitute the values for the mass of the block and net force into the equation:
m1 * a1 = m1 * g - Frictional force

8. Use Newton's second law of motion to relate the acceleration of the block to the net force:
m1 * a1 = m1 * g - (1.20 kg) * 0.3 * g

9. Solve for the acceleration of the block:
a1 = (m1 * g - (1.20 kg) * 0.3 * g) / m1

10. The speed of the blocks can be determined using the final kinetic energy:
KE_final = 1/2 * m1 * v1^2 + 1/2 * m2 * v2^2

11. Substitute the values for the mass of the blocks and the calculated acceleration into the equation:
KE_final = 1/2 * (1.20 kg) * v1^2 + 1/2 * (2.90 kg) * v2^2

12. The initial potential energy of the system can be calculated using the values for the masses and heights:
PE_initial = (1.20 kg) * g * h1 + (2.90 kg) * g * h2

13. Equate the initial potential energy to the final kinetic energy:
(1.20 kg) * g * h1 + (2.90 kg) * g * h2 = 1/2 * (1.20 kg) * v1^2 + 1/2 * (2.90 kg) * v2^2

14. Rearrange the equation to solve for v1^2:
v1^2 = [(1.20 kg) * g * h1 + (2.90 kg) * g * h2 - 1/2 * (2.90 kg) * v2^2] / (1.20 kg)

15. Solve for v1 by taking the square root of both sides of the equation:
v1 = sqrt([(1.20 kg) * g * h1 + (2.90 kg) * g * h2 - 1/2 * (2.90 kg) * v2^2] / (1.20 kg))

16. Substitute the values for the height difference, acceleration due to gravity, and the calculated value for v2 into the equation to calculate the speed of the blocks after moving the given distance.