Grade 11Mechanics

A parallel beam of particles of mass m and velocity v impinges on a wall at an angle θ to its normal. The number of particles per unit volume in the beam is n. If the collision of particles with the wall is elastic, then the pressure exerted by this beam on the wall is:

(A) 2mnv²cosθ (B) 2mnv²cos²θ (C) 2mnvcosθ
(D) 2mnvcos²θ

Samyak Shah

8 Years agoGrade 11

1 Answer

Arun

8 Years ago

Dear Student

I have assumed angle A in place of theta for easy writing.

consider one particle impinging normally on wall with velocity (v cos A)
lets consider momentum conservation (vector in 2 directions). let particle rebound at an angle B after collision
let + x be along beam
normal to wall>
m v cos A + 0 = - m V cos B + 0
parallel to wall
m v sin A + 0 = m V sin B + 0
squaring and adding V = v, A = B [as happens in law of reflection]
------------------------------------
change in momentum normal to wall = m v cos A - (- m v cos A) = 2 m v cos A
n = number of particles/volume
change of momentum/volume = n [2 m v cos A]
--------------------------------------...
if collision takes place in time dt
rate of change of momentum/volume = n [2 m v cos A] / dt
rate of change of momentum = FORCE F exerted = n [2 m v cos A] * [area * dx] / dt
F = n [2 m v cos A] * [area * dx] / dt
Pressure P = F/area = n [2 m v cos A] * [dx] / dt
--------------------------------------...
dx/dt = normal velocity = v cos A
P = n [2 m v cos A] * [v cos A]
P = 2 m n v^2 cos^2 A

hence option B is correct.

Regards

Arun (askIITians forum expert)