A molecule in a gas container hits a horizontal wall with speed 200 m s–1 and angle 30° with the normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic?
Yes; Collision is elastic
The momentum of the gas molecule remains conserved whether the collision is elastic or inelastic.
The gas molecule moves with a velocity of 200 m/s and strikes the stationary wall of the container, rebounding with the same speed.
It shows that the rebound velocity of the wall remains zero. Hence, the total kinetic energy of the molecule remains conserved during the collision. The given collision is an example of an elastic collision.
Let us consider the mass of the molecule be m and that of wall be M. The wall remains at rest due to its large mass. Resolving momentum of the molecule along x-axis and y-axis, we get
The x-component of momentum of molecule
`="mu" cos theta = -m 200 cos 30^@ = -100sqrt3 m`
y component of the molecule
`="mu" sin theta = mxx200xxsin 30^@ = 100m`
Before collision: x-component of total momentum (wall + molecule)
= 0 + (`-100sqrt3` m) = `-100sqrt3 m`
y-component of momentum (wall + molecule) = 0 + 100 m = 100 m
After collision : x-component of the momentum(wall + molecule)
= `0 + m 200 cos 30^@ = 100sqrt3 m`
and y-component = 0 + m 100 sin 30^@ = 100 m`
We find that momentum of the (molecule + wall) system is conserved. The wall has a recoil momentum such that momentum of the wall + momentum of outgoing molecule equals the momentum of the incoming molcule.
Inital kinetic energy (1/2mu^2) is the same as final K.E (`1/2 mv^2`) of the molecule as u = =v = 200 m/s i.e thus the collision is elastic collision