A ball with a mass of #4 kg # and velocity of #3 m/s# collides with a second ball with a mass of #2 kg# and velocity of #- 1 m/s#. If #75%# of the kinetic energy is lost, what are the final velocities of the balls?

Question

A ball with a mass of #4 kg # and velocity of #3  m/s# collides with a second ball with a mass of #2 kg# and velocity of #-  1 m/s#. If #75%# of the kinetic energy is lost, what are the final velocities of the balls?

William Audy · Accepted Answer

The final velocities are #=0.85ms^-1# and #=3.3ms^-1#

We have conservation of momentum

#m_1u_1+m_2u_2=m_1v_1+m_2v_2#

The kinetic energy is

#k(1/2m_1u_1^2+1/2m_2u_2^2)=1/2m_1v_1^2+1/2m_2v_2^2#

Therefore,

#4xx3+2xx(-1)=4v_1+2v_2#

#4v_1+2v_2=10#

#v_2=(5-2v_1)#........................#(1)#

and

#0.25(1/2xx4xx3^2+1/2xx2xx(-1)^2)=1/2xx4xxv_1^2+1/2xx2xxv_2^2#

#4v_1^2+2v_2^2=9.5#

#2v_1^2+v_2^2=4.75#...................#(2)#

Solving for #v_1# and #v_2# in equation s #(1)# and #(2)#

#2v_1^2+((5-2v_1))^2=4.75#

#2v_1^2+4v_1^2-20v_1+25-4.75=0#

#6v_1^2-20v_1-20.25=0#

Solving this quadratic equation in #v_1#

#v_1=(20+-sqrt(20^2-4xx6xx-20.25))/(12)#

#v_1=(40+-sqrt(886))/(12)#

#v_1=(40+-29.8)/(12)#

#v_1=5.81ms^-1# or #v_1=0.85ms^-1#

#v_2=-6.62ms^-1# or #v_2=3.3ms^-1#

Leo Abeyta · Answer

undefined To solve for the final velocities of the balls after the collision, you can use the principle of conservation of momentum and the equation for kinetic energy.

Let $ m_1 = 4 $ kg, $ v_1 = 3 $ m/s, $ m_2 = 2 $ kg, and $ v_2 = -1 $ m/s.

1. Conservation of momentum equation: 
   $$ m_1v_1 + m_2v_2 = m_1v_{1f} + m_2v_{2f} $$

2. Initial kinetic energy of the system:
   $$ KE_{initial} = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 $$

3. Given that 75% of the kinetic energy is lost, the final kinetic energy of the system:
   $$ KE_{final} = 0.25 \times KE_{initial} $$

4. Using the equation for kinetic energy:
   $$ KE_{initial} - KE_{final} = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 $$

Now, you can solve these equations to find the final velocities $ v_{1f} $ and $ v_{2f} $.