A ball with a mass of #6# #kg # and velocity of #4# #ms^-1# collides with a second ball with a mass of #4# #kg# and velocity of #- 5# #ms^-1#. If #75%# of the kinetic energy is lost, what are the final velocities of the balls?
1 Answer
In this instance momentum is conserved. Kinetic energy is not but we know how much was lost.
Explanation:
Before the collision
Momentum:
Kinetic energy:
After the collision
(note that the velocities now are different, even though I have still called the
Momentum:
Kinetic energy:
Now, we know that momentum is conserved, so
We also know that
Now we have two equations in two unknowns, so we can use our tools for 'simultaneous equations' to solve for
Let's divide Equation 1 by 4:
Rearranging to make
We can substitute this expression for
Expanding the brackets:
This is a quadratic equation, which we can solve using the quadratic formula or other methods, but first we need to make the right side equal zero, so we subtract 24.5 from both sides:
Solving it yields:
We can substitute these values back in to our rearranged Equation 1 to find the values of
So
The quadratic equation yields two solutions, corresponding to two possible outcomes of the collision.