Three astronauts in space (in colorful spacesuits) are initially adjacent to each other, floating in space, after which they push off from each other and drift away in different directions. in the figure, the origin of the xy-coordinate system is their original common location and the three circles represent their respective locations one second after pushing apart. the x and y-axis units are meters.

what is the ratio of m_green / m_blue?
what is the ratio of m_red / m_blue?
how far has the center of mass moved after 1 second?

Explanation:

As the three astronauts  they push off from each other, their forces are internal. There is no external force so their momenta are conserved and the sum of their momenta are zero.

By conservation of momentum in the y-direction, m_red * vel_y_red + m_blue * vel_y_blue = 0

After 1 sec, red travels 3 units in y-direction while blue travel -3 units in y-direction. So vel_y_red = 3 and vel_y_blue = -3

m_red*3 + m_blue*(-3) = 0

m_red = m_blue

m_red / m_blue = 1

By conservation of momentum in the x-direction, m_green * vel_x_green + m_red * vel_x_red + m_blue * vel_x_blue = 0

After 1 sec, green travels 3 units in x-direction while red travels -2 units and blue travel -4 units in x-direction. So vel_x_green = 3, vel_x_red = -2 and vel_x_blue = -4

m_green*3 + m_red*(-2) + m_blue*(-4) = 0

m_green*3 = m_red*2 + m_blue*4 = m_blue*6

m_green / m_blue = 6/3 = 2

Because there is no external force, the center of mass has not moved after 1 second.