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## Homework Statement

Two blocks sitting on a frictionless horizontal surface are connected by a light spring. The natural length of the spring is ##d##, and the spring constant is ##k##. The two objects are initially both at rest, with the spring neither stretched nor compressed. At time ##t = 0##, ##m_1## is struck a blow that gives it an initial velocity ##v_0## toward ##m_2##. Taking the initial position of the centre of mass to be zero, use the equations of motion for ##\ddot R## and ##\ddot r## to show that the center of mass and relative position are given by

##R=\frac{m_1v_0}{M}t##, ##r=C+Asin(\omega t+\delta)##

where ##\omega = \sqrt{k/\mu}##. Use the initial condition for ##r## to show ##\delta = 0## and ##C=d##.

## Homework Equations

##m_1\ddot r_1=F##

##m_2\ddot r_2=-F##

##M\ddot R=0##

##\mu\ddot r=F##

##r_1=R+\frac{m_2}{M}r##

##r_2=R-\frac{m_1}{M}r##

## The Attempt at a Solution

So ##\mu\ddot r=F## is the same as ##\mu\ddot r=m_1a_1##. Differentiating to get:

##\mu\dot r=m_1v_1##, and again

##\mu r=m_1r_1##, where ##r_1=v_0t##

Therefore ##\mu r=m_1v_0t##

##\mu r## simplifies to ##MR##. Subbing in and I get required:

##R=\frac{m_1v_0}{M}t##

I have no idea how to get ##r=C+Asin(\omega t+\delta)## however. Any help would be greatly appreciated.