Question #347462

A 1.72 kg ball rolls on a frictionless horizontal surface at 4.19 m/s and collides with a stationary block of mass 1.23 kg which is attached to horizontal spring whose force constant is 1695 N/m. Right after impact, the ball bounces on the opposite direction at 1.59 m/s. To what distance will the spring be compressed?


1
Expert's answer
2022-06-02T10:25:39-0400

Given:

m1=1.72kg,  m2=1.23kgm_1=1.72\:{\rm kg},\; m_2=1.23\:{\rm kg}

v1x=4.19m/s,  v2x=0v_{1x}=4.19\:{\rm m/s},\; v_{2x}=0

v1x=1.59m/s,  k=1695N/mv_{1x}'=-1.59\:{\rm m/s},\; k=1695\:\rm N/m


The law of conservation of momentum says

m1v1x+m2v2x=m1v1x+m2v2xm_1v_{1x}+m_2v_{2x}=m_1v_{1x}'+m_2v_{2x}'

Hence, the velocity of the block after collision with ball

v2x=m1v1x+m2v2xm1v1xm2v_{2x}'=\frac{m_1v_{1x}+m_2v_{2x}-m_1v_{1x}'}{m_2}

v2x=1.721.231.72(1.59)4.19=1.16m/sv_{2x}'=\frac{1.72*1.23-1.72*(-1.59)}{4.19}=1.16\:\rm m/s

The law of conservation of energy says

m2v2x22=k(Δx)22\frac{m_2v_{2x}'^2}{2}=\frac{k(\Delta x)^2}{2}

Thus, the compression of the spring

Δx=m2kv2x=4.1916951.16=5.76102m\Delta x=\sqrt{\frac{m_2}{k}}v_{2x}=\sqrt{\frac{4.19}{1695}}*1.16=5.76*10^{-2}\:\rm m


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