A ball has a kinetic energy of 100 J. What would be the kinetic energy of a ball with twice the mass and half the momentum?

1 Answer
Jacob T.
May 12, 2018

#"KE"_2=12.5color(white)(l)"J"#

Explanation:

There exist a relationship between #"KE"# and #p#, the kinetic energy and momentum of an object of mass #m#.

#"KE"=1/(2m)*p^2#

Proof for this formula

#"L.H.S."="KE"#
#color(white)("L.H.S.")=1/2m*v^2#
#color(white)("L.H.S.")=1/2*(m*v)^2*1/m#
#color(white)("L.H.S.")=1/(2m)*p^2#
#color(white)("L.H.S.")="R.H.S."#

The question states that

  • #p_2=1/2p_1# and
  • #m_2=2m_1#

where #p_1#, #p_2#, #m_1#, and #m_2# the mass and velocity of the first and second ball, respectively,

The kinetic energy of the second ball would be

#"KE"_2=1/(2color(darkblue)(m_2))*color(purple)(p_2)^2#
#color(white)("KE"_2)=1/(2*color(darkblue)(2*m_1))*color(purple)((1/2p_1))^2#
#color(white)("KE"_2)=1/(2color(darkblue)(m_1))*color(purple)(p_1)^2*color(darkblue)(1/2)*(color(purple)(1/2))^2#
#color(white)("KE"_2)=1/8*1/(2color(darkblue)(m_1))*color(purple)(p_1)^2#
#color(white)("KE"_2)=1/8*"KE"_1#
#color(white)("KE"_2)=1/8*100color(white)(l)"J"#
#color(white)("KE"_2)=12.5color(white)(l)"J"#

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