

Conservation of Momentum and Energy
Name: Toby
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
I have been learning about Momentum and Kinetic Energy in
science lessons at school, but there is one thing I really cannot
get my head around. I have been taught that if an object with a
certain momentum (mv) hits another object and all the momentum is
conserved, then the new object will have the same momentum, so if
the new object had half the mass of the first it would travel at
double the speed, e.g. 100kg object travelling at 10m/s hits a 50kg
object, so the 50kg object will go at 20m/s (50 * 20 = 100 * 10).
What confuses me is when I try to work out the Kinetic Energy of
the two objects. If the 100kg object is travelling at 10m/s then it
has kinetic energy of 5000J (100/2 * 10^2), but if it hits a 50kg
object and all the momentum is conserved then the new object will
have kinetic energy of: 50/2 * 20^2 = 10000J. I was under the
impression that energy could not be created or destroyed, so how
does this 100kg object travelling through a vacuum with 5000J of
energy suddenly hit this 50kg object and produce 10000J of energy,
where does the extra energy come from?
Replies:
Hi Toby
You are right in questioning the idea that both momentum and energy
must be conserved. In an elastic collision, both are conserved. In the
example you gave, the total momentum was 1000 kg m/s. That must be the
total momentum after the collision. Your example assumes that all the
momentum will be transferred to the second object. That will happen
only if both objects are the same mass. In your example, the second
object is half the mass of the first. So what will happen is the first
object will collide with the stationary second object, the second will
take off, but the first will not stop, it will just slow down. In that
way, the total momentum is conserved, along with total energy. It
turns out that the relative speed of two objects after an elastic
collision has the same magnitude as before the collision regardless of
the masses involved.
When a 100 kg mass traveling at 10 m/s collides with a 50 kg
stationary mass, the 100 kg object will continue on traveling at 3.33
m/s, while the 50 kg mass takes off at 13.33 m/s. You will see that
total momentum and total energy are conserved.
Hope this helped.
Bob Froehlich
Toby,
You are quite right, BOTH momentum and energy are conserved. As a
result, the first scenario you describe cannot happen, because it
would violate the laws of physics. You see, not all of the momentum
of the first (initially moving) object will be transferred to the
second (initially stationary) object unless both objects have exactly
the same mass (and the collision is deadon). in an elastic collision,
both momentum and kinetic energy are conserved. In fact, knowing that
makes it possible to figure out exactly what the velocities are after
the collision.
In an inelastic collision, in which the colliding objects cling together
afterwards, kinetic energy is not explicitly conserved. The energy is
not lost, however; it becomes thermal energy (higher temperature) in
the two objects.
No doubt you will learn much more about this as you go along. It is
fascinating stuff, and I hope you will enjoy it as much as I do.
Richard Barrans
Department of Physics and Astronomy
University of Wyoming
Toby,
The second object will not necessarily get all of the momentum. Some
may stay in the first object. One such example is when the two objects
stick together. Another is when the first object bounces back from that
which it hits. Remember that direction counts for momentum. If two
objects are traveling in opposite directions, their momentums have
opposite signs.
As for energy, not all energy is kinetic energy. When objects collide,
they can lose kinetic energy by producing sound, by damaging each other,
or even by producing heat. Energy is still conserved, but it takes a
form we cannot easily measure. Some collisions, called elastic
collisions, do not lose any kinetic energy. Such collisions are very
quiet, often involving very little contact.
If the colliding objects stick together, kinetic energy is always lost.
It takes energy to make the objects stick together. This often involves
reshaping, such as bending Velcro hooks or flattening a piece of clay.
It can involve destruction, such as when two cars have a headon
collision.
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
In perfectly elastic collisions, both kinetic energy and momentum are
conserved.
Using your example, with a 100kg object traveling at 10m/s, its momentum is
1,000 kgm/s. If it then hits a 50kg object at rest, the resultant collision
must fulfill both of the above constraints (momentum and kinetic energy
conserved). The solution you propose to the problem  that the 50 kg object
will go at 20m/s and the 100kg object will be at rest  does not fulfill the
kinetic energy constraint, and so is not correct. That solution meets the
conservation of momentum constraint, but not the conservation of kinetic
energy constraint.
If you solve the equations (m11 * v11 ^2 + m21 * v21 ^2) = (m12 * v12 ^2 +
m22 * v22 ^2) and (m11 * v11 + m21 * v21) = (m21 * v21 + m22 * v22)
(where the first number is the object, and the second number is the time 
e.g. mass 1 at the start is m11 and mass 1 and the end is m12)
You know m1 and m2 (they don't change) and you set v11 and v21. You have two
equations and two unknowns (v12 and v22) so you can solve this system
directly algebraically.
Also, here is a great site that goes over examples:
http://hyperphysics.phyastr.gsu.edu/hbase/colsta.html
It has a form to do these calculations for you too!
Hope this helps,
Burr Zimmerman
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Update: June 2012

