a. A stone is dropped at t = 0 s. A second stone, with a mass 2.0 times that of the first, is dropped from the same point at t = 0.25 s. How far from the release point is the center of mass of the two stones at t = 0.52 s? Assume that neither stone has yet reached the ground.

b. What is the speed of the center of mass of the two stone system at that time?

x1 = .5 (9.89).52^2 = 1.32

x2 = .5 (9.8).27^2 = .36
Distance between is .96 and center of mass 1/3 distance from stone 2 = .31.
COM is therefore .36 + .31 = .67
b. v^2 = 2ax = 2(9.8).67 = 13.1

I do not know

To find the distance and speed of the center of mass of the two-stone system at a given time, we can apply the concept of center of mass and use the equations of motion.

a. Distance of the center of mass:
The center of mass of a two-object system can be calculated using the formula:

X_cm = (m1*x1 + m2*x2) / (m1 + m2)

Where X_cm is the position of the center of mass, m1 and m2 are the masses of the respective objects, and x1 and x2 are their respective positions.

Let's denote the position of the first stone at time t = 0 as x1, and the position of the second stone at t = 0.25 s as x2.

Since the first stone is dropped at t = 0, its position can be given as:
x1 = 0.5 * g * t^2

Where g is the acceleration due to gravity (approximately 9.8 m/s^2) and t is the time in seconds.

For the second stone dropped at t = 0.25 s, its position can be calculated as:
x2 = 0.5 * g * (t - 0.25)^2

Now, substitute the values into the center of mass formula:
X_cm = (m1*x1 + m2*x2) / (m1 + m2)

Given that the mass of the second stone (m2) is 2.0 times that of the first stone (m1), we have:
m2 = 2.0 * m1

Therefore, the center of mass distance at t = 0.52 s is given by:
X_cm = (m1*x1 + 2.0*m1*x2) / (m1 + 2.0*m1)

Simplifying this expression will yield the answer.

b. Speed of the center of mass:
To find the speed of the center of mass at a given time, we can differentiate the position equation with respect to time, apply the same center of mass formula, and then find the magnitude of the velocity vector.

V_cm = (m1*v1 + m2*v2) / (m1 + m2)

Where V_cm is the velocity of the center of mass, and v1 and v2 are the velocities of the objects at their respective positions.

Differentiate the position equations with respect to time to get the velocities:
v1 = g * t
v2 = g * (t - 0.25)

Substituting these values into the center of mass velocity formula will give the answer.

Note: Make sure to double-check all the calculations and units to get accurate results.