The two related concepts to solve this problem are those concerning the conservation of the Moment and the simple harmonic movement.
By conservation of the moment we can find the speed, while by the simple harmonic movement the Period.
By definition the conservation of momentum is given by
[tex]v_1m_1+v_2m_2 = (m_1+m_2)v_f[/tex]
Where,
[tex]v_{1,2}[/tex] = Velocity of each object
[tex]v_f[/tex] = Final velocity
[tex]m_{1,2}[/tex] = Mass of each object
Since the first body does not have speed, the equation would be subject to,
[tex]v_2m_2 = (m_1+m_2)v_f[/tex]
[tex]250*120=(500+250)*v_f[/tex]
[tex]v_f=40 cm/s[/tex]
From the equations of the harmonic movement we can relate the final velocity with the amplitude and angular frequency and the amplitude with the velocity. In mathematical terms,
[tex]V=Aw[/tex]
Where A is the amplitude and W is the angular frequency, which is also defined as,
[tex]w= \sqrt{(\frac{K}{m})}[/tex]
Replacing this function and clearing for the amplitude we have to,
[tex]A=V*\sqrt{\frac{m}{K}}[/tex]
[tex]A=40*\sqrt{\frac{(0.5+0.25)}{10}}[/tex]
[tex]A=10.95 cm[/tex]
Therefore the amplitude is 10.95cm
At the same time through the angular frequency we can find the period under the definition,
[tex]T =\frac{2\pi}{w}[/tex]
[tex]T =\frac{2\pi}{\sqrt{\frac{K}{m}}}[/tex]
[tex]T =2\pi*\sqrt{\frac{m}{K}}[/tex]
[tex]T=2\pi*\sqrt{0.75}{10}[/tex]
[tex]T =1.72 s[/tex]
