A rail-road car is moving towards right with acceleration a. A man accelerating toward left with an acceleration of magnitude `a//3` w.r.t to car. A dog of mass m is following man A with an acceleration `a//3` relative to the car. Observer B on ground is observing the dog and man A. Find the
(a) net force experienced by the dog as seen by observer B standing on ground.
(b) rate of change of linear momentum of the dog relative to the man A moving on trolley.
(c ) pseudo-free on the dog as seen from man A.

Explanation

(a) The net force acting on dog, as seen from observer 'B' on ground is
`vec(F)=m vec(a)_("dog,ground")=m vec(a)_(d)` …(ii)
Acceleration of dog = Acceleration of dog w.r.t car + Acceleration of car
`implies vec(a)_(d) = vec(a)_(d,c) + vec(a)_(c) =-a/3 hat(i)+a hat(i) =(2a)/(3)hat (i)`
Hence, from (i) , `vec(F) =(2ma)/(3) hat(i)`
(b) The rate of change of momentum of the dog relative to the man `A(d vec(p))/(dt) = mvec(a)_(d,A)` ...(ii)
where `vec(a)_(d,A) = vec(a)_(d,C)=vec(a)_(A,C)=-(a)/(3) hat(i) -((-a)/(3)hat(i))=0`
Hence from (ii) `(dvec(p))/(dt)=0`
(c ) Pseudo-force acting on dog viewed by the man A is
`vec(F)_(pseudo)=-m vec(a)_(A,ground)`
where `vec(a)_(A,ground)=vec(a)_(A,C)+vec(a)_(C ) = -(a)/(3) hat(i) +a hat(i)=(2a)/(3) hat(i)`
This gives `vec(F)_(pseudo)=-(2ma)/(3)hat(i)`.