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Two Body system: Motion of two masses attached by spring

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Two masses attached with a spring and one mass is projected vertically upward. Position of the two masses as a function of time are calculated.

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Here it is not told that m2 is placed on a horizontal surface and thus we will consider that at t = 0 m1 is moving with velocity v, m2 is at rest and just released.
Let at the initial position of center of mass of the system is the origin.
As initially velocity v is imparted to m1 only the initial velocity of center of mass of this two body system is given by

As the only external force on the system is its weight (m1 +m2)*g downwards, the position of center of mass at any subsequent time is given by

Or
Now let the extension in the spring at time t is
Then the tension in the spring is given by F = k
Now the motion of the two bodies relative to their center of mass is due to internal forces i.e. the tension in the string. Let at time t the position of the two bodies are y1 and y2 relative to their ...

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