Momentum and Kinetic Energy
If a particle that is moving has the same momentum
and the same kinetic energy as another particle, must their masses
and velocities be equal?
I do not understand the properties of two particles that have the
same momentum, but different kinetic energies.
Momentum is mass times velocity and kinetic energy is one-half mass
times velocity squared. If I have a ball with mass M traveling at
speed 2V, and a ball of mass 2M traveling at speed V, they both have
the same momentum, 2MV. Kinetic energy, however, is a different
story. The first ball has kinetic energy of one-half M times the
quantity 2V squared or one-half M x 4 V-squared or 2MV-squared. The
second ball has kinetic energy one-half 2M x V-squared or MV-squared.
The first ball has twice the kinetic energy of the first, even though
they have the same momentum.
This explains why when two billiard balls collide with a moving ball
striking a stationary ball head-on, the moving ball stops and the
stationary ball starts moving at the first ball's velocity. In this
case, both momentum and kinetic energy were conserved. If the first
ball struck the second and they both took off at half the velocity of
the original ball, momentum would be still be conserved but kinetic
energy would not.
Hope this helps.
Momentum is m*v, and kinetic energy is m*v*v/2, so if momenta and energies
are the same, we have:
1) m1*v1 = m2*v2
2) m1*v1*v1 = m2*v2*v2
using (1) in (2) yields
m1*v1*v1 = (m1*v1)*v2 -> v1 = v2
using this in (1) shows that m1 = m2
So, yes, if a particle has the same momentum and the same kinetic energy
as another particle, their masses and velocities must be equal
But having the same momentum does not by itself imply having the same
energy. A heavy particle moving slowly can have the same momentum as
a light particle moving swiftly.
Advanced Photon Source, Argonne National Lab.
First, energy and momentum are very different properties.
An object takes time to stop moving. The momentum of an object can be
describes as the amount of force required to stop the object in one second.
When a constant force is applied to an object, it is the force multiplied by
the time over which it is applied that yields the change of momentum of the
particle. Also, momentum is a vector: it has a direction to it. Momentum
points in the direction of an object's velocity.
An object continues to travel while its velocity drops to zero. The kinetic
energy of an object can be described as the amount of force required to stop
the object over a distance of one meter. When a constant force is applied
to an object, it is the force multiplied by distance traveled along the SAME
AXIS as the force that determines the change of kinetic energy. Kinetic
energy is a scalar: it has no direction.
An object changing direction but neither speeding up nor slowing down is an
example of changing momentum but not changing kinetic energy. If the object
does speed up or slow down, both momentum and kinetic energy will change.
For example, consider throwing a rock upward at a certain speed. If you
double the rock's initial speed, the rock will require twice the time and
four times the distance to reach zero speed. Thus, the rock with the
doubled speed has twice the momentum and four times the kinetic energy.
Dr. Ken Mellendorf
Illinois Central College
Yes as a little algebra should convince you, if two particles have
the same momentum and the same kinetic energy, their masses and
speeds must be the same.
Just take the equation m1 v1^2 = m2 v2^2 (equal kinetic energies)
and divide by m1 v1 = m2 v2 (equal momenta). You get v1 = v2. Put
that in m1 v1 = m2 v2 and you get m1 = m2.
To understand two particles that have the same momentum but
different kinetic energies, write the kinetic energy as
p^2/(2m). So if two particles have the same momentum (p), their
kinetic energies are proportional to the inverse of their masses.
Best, Dick Plano, Professor of Physics emeritus, Rutgers University
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Update: June 2012