Solving Simple Harmonic Motion Question

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acmmanoj
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I am having a question and tries to solve a problem for days. Consider general SHM. When the particle reaches to the maximum displacement, a if a velocity U is given to the particle towards to the center of SHM, (keeping the same force mω^2x)

1. What would happen to the SHM...is it same or can i use same equations or should i derive equation again

Should i consider this as new SHM ( X=a when t=0) or should i continue from same SHM (X=0 when T=0)

2. if i derive again, which point should i considered as center, what will happen to the displacement and maximum displacement...is it same or difference

3. when tries to get equation of motion as x=asin(ωt) from intergartion it produces a very complex equation. (in intergration i took, when x=a , v=u and x=a t=0)
 
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