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After a block and marble collide, the block executes SHM

A spring with negligible mass, and force constant k= 60 nt/m, has one end fixed, and the other end attached to a stationary block of mass M= 2.0 kg on a horizontal, frictionless table. At time t=0, a ball of mass m= .25 kg, with horizontal speed of Vb= 50 m/sec to the right before impact, hits the block and bounces off, then moving with speed Va= 40 m/sec to the left. See ATTACHMENT for a diagram with vectors and action sequence. After impact, the block executes SHM of amplitude Xm.
Find x(t) for the SHM of the block, including Xm in meters, angular frequency w in rad/sec, and initial phase Q in radians.


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Physics notes:
A. In a collision, momentum is conserved, (but kinetic energy is not). Therefore we must equate total momentum before to that after the collision to get (maximum) velocity V of the block at x= 0, at t=0 as SHM starts.
B. After the collision, energy is conserved, as the block's kinetic energy is transformed into potential energy of the spring as the block slows down and the spring compresses.
C. From the impact point to the point where the block is momentarily brouht to rest is the amplitude Xm of the SHM.
D. The initial conditions of the SHM are: at t=0, x= 0 ...

Solution Summary

A spring with negligible mass with force constants are determined. A diagram with vectors and action sequences are determined.