A simple Atwood's machine uses two masses, m1 and m2. Starting from rest, the speed of the two masses is 7.0 m/s at the end of 5.0 s. At that instant, the kinetic energy of the system is 100 J and each mass has moved a distance of 17.5 m. Determine the values of m1 and m2.

To solve this problem, we can use the principles of conservation of energy and Newton's second law.

First, let's use the principle of conservation of energy to determine the total potential energy at the beginning and the end of the 5.0 s interval.

At the beginning, both masses are at rest, so their kinetic energy is zero. The total potential energy at this point is due to the height difference between the masses and can be calculated as:

Initial Potential Energy (PE_initial) = m1 * g * h

where m1 is the mass of the first mass, g is the acceleration due to gravity, and h is the height difference between the masses.

At the end of the 5.0 s interval, the kinetic energy is given as 100 J. Therefore, the total potential energy at this point is:

Final Potential Energy (PE_final) = m1 * g * h + m2 * g * h

Next, we use Newton's second law to relate the net force to the acceleration of the system. The net force is the difference between the two tension forces acting on each mass.

Net Force = Tension on m2 - Tension on m1

Applying Newton's second law to each mass:

m1 * a = Tension on m1
m2 * a = Tension on m2

Since the acceleration of the system is the same for both masses, we can write:

m1 * a = m2 * a

Simplifying, we find:

m1 = m2

Now, let's substitute this into the equation for the net force:

Net Force = m2 * a - m2 * a
Net Force = 0

Since there is no net force acting on the system, the tension forces must cancel each other out.

Now, let's determine the acceleration of the system. We know that the distance moved by each mass is 17.5 m. Using the kinematic equation:

distance = initial velocity * time + 0.5 * acceleration * time^2

For each mass, we have:

17.5 = 0 * 5 + 0.5 * a * 5^2
17.5 = 12.5 * a

Solving for acceleration (a):

a = 17.5 / 12.5
a = 1.4 m/s^2

Now, we can use this acceleration to solve for the masses. Since we already determined that m1 = m2, we will use the variable m for both masses.

m1 = m2 = m

Since the net force is zero, we can set up the equation using Newton's second law:

m * a = m * g + m * g

Substituting the value of acceleration (a) and acceleration due to gravity (g):

m * 1.4 = m * 9.8 + m * 9.8
m * 1.4 = m * 19.6

Dividing both sides by m:

1.4 = 19.6

This is not possible, so there must be an error in the problem statement or the calculations. Please check the given information and re-calculate if necessary.