A ball with a mass of # 3# # kg# is rolling at #18 # #ms^-1# and elastically collides with a resting ball with a mass of #9 # #kg#. What are the post-collision velocities of the balls?

1 Answer
Feb 24, 2016

Answer:

In an elastic collision, both momentum and kinetic energy are conserved. If we call the #3# #kg# ball 1 and the #9# #kg# ball 2, their final velocities are #v_1=-14.4# and #v_2=10.8# #ms^-1#.

Explanation:

Initial momentum:

#p=m_1v_1+m_2v_2=3*18+9*0=54# #kgms^-1#

Initial kinetic energy:

#E_k=1/2m_1v_1^2+1/2m_2v_2^2=1/2*3*18^2+1/2*9*0^2=486# #J#

Because this is an elastic collision, both will be conserved:

Final momentum:

#p=54=m_1v_1+m_2v_2=3v_1+9v_2# - call this #(1)#

Final kinetic energy:

#E_k=486=1/2m_1v_1^2+1/2m_2v_2^2=1/2*3*v_1^2+1/2*9*v_2^2# - call this #(2)#

We now have two equations, #(1)# and #(2)#, in two unknowns.

Let's double #(2)#, just for neatness:

#972=3v_1^2+9v_2^2#

We can find a value for #v_1# by rearranging #(1)# and then substitute that into this revised #(2)# so that we are working with only one variable:

#54=3v_1+9v_2#

#v_1=(54-9v_2)/3#

Then:

#972=3((54-9v_2)/3)^2+9v_2^2=(54-9v_2)^2/3+9v_2^2#

#972=(2916-972v_2+81v_2^2)/3+9v_2^2#

Let's multiply through by 3 for neatness:

#2916=2916-972v^2+81v_2^2+9v_2^2#

Rearranging:

#90v_2^2-972v_2=0#

Solve using the quadratic formula or otherwise, and you get:

#v_2=(972+-sqrt(944784-0))/180=(972+-972)/180#

There are actually two roots, therefore, #0# and #10.8# #ms^-1#.

That is, the second ball, with a mass of #9# #kg# can be either stationary or moving at #10.8# #ms^-1#.

We should solve #(1)# with both of these to find the possible values of #v_1#:

If #v_2=0#, #v_1=54/3=18ms^-1#

But wait: this is just the condition before the collision! The #3# #kg# ball has a velocity of #18# #ms^-1# and the #9# #kg# ball is stationary!

If #v_2=10.8#, #v_1=-14.4# #ms^-1#.

The minus sign indicates that this velocity is in the opposite direction to the original velocity.

The #3# #kg# ball approaches at #18# #ms^-1# then collides and bounces backward at #14.4# #ms^-1#, propelling the #9# #kg# ball forward at #10.8# #ms^-1#.