Centre of Gravity in a Two-Object System: Calculating with Mass and Height

[cfc101]

Homework Statement

A small ball of mass m is dropped at a height of 50 metres. At the same time, a second ball with twice the mass of the first one is launched at an initial velocity of 10m/s from the ground. Where is the centre of gravity in this system?

Homework Equations

The Attempt at a Solution

I don't really know where to begin on this one, although i know that the cener of mass will probably lie closer to the ground because of heavier ball. All help is appreciated, thanks in advance.

 

[Astronuc]

How does one define center of mass, given the position of two masses in the y-direction?

What is happening with the position of those masses?

See if this helps - http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html

Remember if something is falling and it's position is y(t) from some initial height h from the ground, then it's height or elevation is given by h-y(t).

 

[cfc101]

How does one define center of mass, given the position of two masses in the y-direction?

What is happening with the position of those masses?

See if this helps - http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html

Remember if something is falling and it's position is y(t) from some initial height h from the ground, then it's height or elevation is given by h-y(t).

Well, since this problem only deals with the y axis, the center of mass is 1/3h off the ground (if stationary, using the mass ratio), however, I don't really understand how velocity is factored into all of this

 

[Astronuc]

Well, since this problem only deals with the y axis, the center of mass is 1/3h off the ground (if stationary, using the mass ratio), however, I don't really understand how velocity is factored into all of this

The positions (elevations) of each mass are changing with time.

y_cm(t) = (y₁(t)m₁+y₂(t)m₂)/(m₁+m₂), so determine y_i(t).

What are the positions y₁(t) and y₂(t) as functions of inital position, velocity, acceleration and time.

 

[cfc101]

The positions (elevations) of each mass are changing with time.

y_cm(t) = (y₁(t)m₁+y₂(t)m₂)/(m₁+m₂), so determine y_i(t).

What are the positions y₁(t) and y₂(t) as functions of inital position, velocity, acceleration and time.

h = h0 + Vot + 1/2 t^2
so for the first mass at the top

h1 = 50 -4.9t^2
mass at bottom

h2 = 10t -4.9t^2

To get the center of mass, would i need to add these two equations?

Cm = -9.8t^2 + 10t + 50 -----> -4.9t + 5t + 25

I have no idea what to do from here

 

[Astronuc]

Referring to this - http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html#cm

h_cm(t) = (m1*h1(t)+m2h2(t))/(m1+m2) where the distances h1 and h2 are with respect to the ground.