A ball with a mass of # 5 kg# is rolling at #8 m/s# and elastically collides with a resting ball with a mass of #2 kg#. What are the post-collision velocities of the balls?

Question

A ball with a mass of # 5    kg# is rolling at #8    m/s# and elastically collides with a resting ball with a mass of #2  kg#. What are the post-collision velocities of the balls?

Olivia Askew · Accepted Answer

#v_1^'=24/7" " m/s#
#v_2=80/7" "m/s# #m_1=5" " kg# #v_1=8" "m/s# #m_2=2" " kg# #v_2=0# #"momentum before collision:"# #P_b=m_1*v_1+m_2*v_2# #P_b=5*8+2*0# #P_b=40+0# #P_b=40 " "kg*m/s# #"momentum after collision:"# #P_a=m_1*v_1^'+m_2*v_2^'# #P_a=5*v_1^'+2*v_2^'# #P_b=P_a" conservation of momentum"# #40=5*v_1^'+2*v_2^'" (1)"# #m_1*v_1+m_2*v_2=m_1*v_1^'+m_2*v_2^' " (3)"# #1/2m_1*v_1^2+1/2*m_2*v_2^2=1/2*m_1*v_1^('2)+1/2*m_2*v_2^(2)'" (4)"# #"we obtain the equation of "v_1+v_1^'=v_2+v_2^'" ;"# #"if we arrange together the equation (3) and (4)"# #8+v_1^'=0+v_2^'# #v_2=8+v_1^'" "(5)# #"now; let's use the equation of (1)"# #40=5*v_1^'+2*(8+v_1^')# #40=5.v_1^'+16+2*v_1^'# #40-16=7*v_1^'# #24=7*v_1^'# #v_1^'=24/7" " m/s# #"now;let's use the equation of (5)"# #v_2=8+24/7# #v_2=(56+24)/7# #v_2=80/7" "m/s# #"is solution true ?"# #cancel(1/2)m_1*v_1^2+cancel(1/2)*m_2*v_2^2=cancel(1/2)*m_1*v_1^('2)+cancel(1/2)*m_2*v_2^(2)'# #5*8^2+0=5*(24/7)^2+2*(80/7)^2# #5*64=5*576/49+2*6400/49# #320=(2880+12800)/49# #320=15680/49# #color(green)(320=320" True")#

Carson Alexander · Answer

undefined To calculate the post-collision velocities, you can use the conservation of linear momentum. The equation is:

$$ m_1 \cdot v_{1i} + m_2 \cdot v_{2i} = m_1 \cdot v_{1f} + m_2 \cdot v_{2f} $$

Substitute the given values:

$$ 5 \, \text{kg} \cdot 8 \, \text{m/s} + 2 \, \text{kg} \cdot 0 \, \text{m/s} = 5 \, \text{kg} \cdot v_{1f} + 2 \, \text{kg} \cdot v_{2f} $$

Solve for $v_{1f}$ and $v_{2f}$.