A ball with a mass of #5 kg # and velocity of #6 m/s# collides with a second ball with a mass of #3 kg# and velocity of #- 7 m/s#. If #40%# of the kinetic energy is lost, what are the final velocities of the balls?

Question

A ball with a mass of #5 kg # and velocity of #6  m/s# collides with a second ball with a mass of #3 kg# and velocity of #-  7 m/s#. If #40%# of the kinetic energy is lost, what are the final velocities of the balls?

Nathan Ashcroft · Accepted Answer

The final velocities are #=-3.18ms^-1# and #=8.27ms^-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,

#5xx6+3xx(-7)=5v_1+3v_2#

#5v_1+3v_2=9#

#v_2=((9-5v_1))/3#........................#(1)#

and

#0.60(1/2xx5xx6^2+1/2xx3xx(-7)^2)=1/2xx5xxv_1^2+1/2xx3xxv_2^2#

#5v_1^2+3v_2^2=196.2#...................#(2)#

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

#5v_1^2+3(((9-5v_1))/3)^2=196.2#

#15v_1^2+25v_1^2-90v_1+81-196.2=0#

#40v_1^2-90v_1-115.2=0#

#20v_1^2-45v_1-57.6=0#

Solving this quadratic equation in #v_1#

#v_1=(-45+-sqrt(45^2+4xx20xx57.6))/(40)#

#v_1=(-45+-sqrt(6633))/(40)#

#v_1=(-45+-81.44)/(40)#

#v_1=-3.16ms^-1# or #v_1=0.91ms^-1#

#v_2=8.27ms^-1# or #v_2=-1.48ms^-1#

Emily Abbate · Answer

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

First, we calculate the initial momentum of the system:

$p_{\text{initial}} = m_1v_1 + m_2v_2$

Next, we use the conservation of momentum to find the final momentum of the system:

$p_{\text{final}} = p_{\text{initial}}$

Then, we calculate the initial kinetic energy of the system:

$KE_{\text{initial}} = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2$

Given that 40% of the kinetic energy is lost, the final kinetic energy can be expressed as:

$KE_{\text{final}} = (1 - 0.40) \times KE_{\text{initial}}$

Finally, we use the final kinetic energy to find the final velocities of the balls using the equation for kinetic energy:

$KE_{\text{final}} = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2$

By solving these equations simultaneously, we can determine the final velocities of the balls.