A stationary ball of radius `R` is irradiated by a parallel stream of particles whose radius is `r`. Assuming the collision of a particle and the ball to be elastic find:
(a) the deflection angle `theta` of a particel as a function of its aiming parameter `b`:
(b) The fraction of a particles which after a collision with the ball are scattered into the angular interval between `theta` and `theta+ dtheta`:
(c )the probability of a particle to be deflected, after a collision with the ball, into the front hemisphere `( thetalt(pi)/(2))`

Explanation

It is implied that the ball is too heavy to recoil.
(a) The trajectroy of the particle is symmertical about the radius vector through the point of impact. It is clear from the diagram that
`theta= pi-2 varphi or varphi=(pi)/(2)-(theta)/(2)`.
Also `b=(R+r) sin varphi= (R+r)"cos"(theta)/(2)`.
(b) With `b` defined above, the fraction of particle scattered between `theta` and `theta+d theta`(or the probability of the same) is
`dP=(|2 pibdb|)/(pi(R+r)^(2))=(1)/(2) sin theta d theta`
(c ) This is
`P= int_(0)^(pi//2)(1)/(2)sin thetad theta=(1)/(2)int_(-1)^(0)d(-cos theta)=(1)/(2)`