While
riddled with tedious algebra, elastic collision problems are quite simple to
solve given the correct procedure. In
fact, there are two ways to solve them: the LAB frame and the CM frame. The LAB frame refers to the point of view of
an observer in a stationary reference frame while the CM frame, or center of
mass frame, refers to the point of view of an observer from a frame moving with
the center of mass of the colliding particles.
While it is necessary to know something about what happens after the
collision in either case, approaching the scenario from the CM frame eliminates
linear momentum and does not require solving a system of equations to determine
the motion of the system. In this blog,
I will demonstrate the process for solving a system of colliding particles in
the LAB frame.
When
taking the LAB frame approach, one must simply formulate and solve the system
of equations describing the conservation of energy and linear momentum of the
collision. Consider the collision below.
For this example, we will assume particles of initial masses
m1 and m2 and initial velocities u1 and u2. While these
velocities can be vectors in any direction, we will assume u1 and u2
are only along the x axis. Furthermore, as I previously mentioned, we must
know at least one thing about what happens after the collision. Thus, for this example, we will assume that
particle 1 deflects at an angle of θ1. Next, we will construct our system of
conservation equations. As we know
energy and linear momentum must be conserved, we know
and
where v1
and v2 represent the
magnitude of the respective particles’ velocities after the collision and θ2 is the scatter angle of
the second particle. The first equation
is the conservation of kinetic energy while the second and third equations are
the conservation of linear momentum in the x
and y coordinates, respectively. Finally, to determine the two particles’
velocities and particle 2’s scatter angle after the collision, we must simply
solve the system of conservation energies.
Applying this technique for the case in which
and
we find that
and
In the next post, I will demonstrate how to solve this same scenario from the CM frame.
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