In an elastic collision,
the sum of the colliding objects’ kinetic energies
is the same before and after the collision.
In contrast, in an inelastic collision,
some of that energy is converted into another form,
for example heat.
In 1D, not only the kinetic energy is conserved, but also the total momentum.
Let and be the initial velocities of objects 1 and 2,
and and their velocities afterwards:
After some rearranging,
these two equations can be written as follows:
Using the first equation to replace
with in the second:
Dividing out the common factors
then leads us to a simplified system of equations:
Note that the first relation is equivalent to ,
meaning that the objects’ relative velocity
is reversed by the collision.
Moving on, we replace in the second equation:
Dividing by ,
and going through the same process for ,
we arrive at:
To analyze this result,
for practicality, we simplify it by setting .
In that case:
How much of its energy and momentum does object 1 transfer to object 2?
The following ratios compare and to quantify the transfer:
If , both ratios reduce to ,
meaning that all energy and momentum is transferred,
and object 1 is at rest after the collision.
Newton’s cradle is an example of this.
If , object 1 simply bounces off object 2,
barely transferring any energy.
Object 2 ends up with twice object 1’s momentum,
but is very small and thus negligible:
If , object 1 barely notices the collision,
so not much is transferred to object 2:
- M. Salewski, A.H. Nielsen,
Plasma physics: lecture notes,