A ball with a mass of # 6 kg# is rolling at #2 m/s# and elastically collides with a resting ball with a mass of # 8 kg#. What are the post-collision velocities of the balls?

Question

A ball with a mass of # 6 kg# is rolling at #2  m/s# and elastically collides with a resting ball with a mass of #   8    kg#. What are the post-collision velocities of the balls?

Charles Aeschliman · Accepted Answer

The post collision velocities of the balls are #=-2/7ms^-1# and #=12/7ms^-1#

As the collision is elastic, there is conservation of momentum and conservation of kinetic energy

#m_1u_1+m_2u_2=m_1v_1+m_2v_2#

#1/2m_1u_1^2+1/2m_2u_2^2=1/2m_1v_1^2+1/2m_2v_2^2#

The mass the first ball is #m_1=6kg#

The velocity of the first ball before the collision is #u_1=2ms^-1#

The mass of the second ball is #m_2=8kg#

The velocity of the second ball before the collision is #u_2=0ms^-1#

The velocity of the first ball after the collision is #=v_1#

The velocity of the second ball after the collision is #=v_2#

#6*2+8*0=6v_1+8v_2#

#3v_1+4v_2=6#

#v_1=(6-4v_2)/3#.....................#(1)#

#1/2*6*2^2+1/2*8*0=1/2*6*v_1^2+1/2*8*v_2^2#

#6v_1^2+8v_2^2=24#

#3v_1^2+4v_2^2=12#...........................#(2)#

From equations #(1)# and #(2)#, calculate #v1# and #v_2#

#3*((6-4v_2)/3)^2+4v_2^2=12#

#36-48v_2+16v_2^2+12v_2^2=36#

#28v_2^2-48v_2=0#

#4v_2(7v_2-12)=0#

#v_2=0# or #v_2=12/7#

#v_1=2# or #v_1=-2/7#

Isabella Alvarez · Answer

undefined To find the post-collision velocities of the balls, you can use the conservation of momentum and kinetic energy principles.

1. Calculate the total momentum before the collision.
2. Apply conservation of momentum to find the velocities after the collision.
3. Use the conservation of kinetic energy to solve for the velocities.

Given:
- Mass of ball 1 (rolling): m1 = 6 kg
- Initial velocity of ball 1: v1 = 2 m/s
- Mass of ball 2 (resting): m2 = 8 kg

1. Total momentum before the collision:
   p_initial = m1 * v1 + m2 * v2 (since ball 2 is at rest initially)

2. Apply conservation of momentum:
   p_initial = p_final
   m1 * v1 + m2 * v2_initial = m1 * v1_final + m2 * v2_final

3. Use conservation of kinetic energy:
   (1/2) * m1 * v1^2 + (1/2) * m2 * v2_initial^2 = (1/2) * m1 * v1_final^2 + (1/2) * m2 * v2_final^2

Solve these equations simultaneously to find the velocities of the balls after the collision.