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
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"