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In a cyclinder a heavy piston is moving ...

In a cyclinder a heavy piston is moving with speed v as shown diagram and gas is filled inside it. A gas molecule is moving with speed `v_(0)` towards moving piston. Then which of the following is correct (Assume `v lt lt lt lt v_(0)(Delta l)/(l)` and coillision is elastic)

A

change in speed after collision is 2V

B

change is speed after collision is `2v_(0)(Deltal)/(l)`

C

rate of collision is `(V)/(l)`

D

When piston is at `(l)/(2)` its kinetic energy will be four times

Text Solution

Verified by Experts

The correct Answer is:
A

`Therefore change in speed `=(2V+V_(0)-V_(0))`
Since the speed of piston is very small thus approximate time between successive collisoins will be
`t=(2l)/(V_(0))` (time for the molcule to travel 2l distance)
`:.` frequency /rate of collision =`1/t=(V_(0))/(2l)`
The speed of the molcule after `1^(st)` collision with piston =`V_(0)+2v`
`:.` after `2^(nd)` collision =`v_(0)+4V`
`:.` after `n^(th)` collison `=v_(0)+2nv =v_(f)`
The collision in time `Deltat=(l//2/v` (during whcih pisiton gets closer to the wall by `l/2`) is
`n=fDeltat=(v_(0))/(2l)xxl/(2v)=(v_(0))/(4v)`
`:. 2nv=(v_(0))/2`
`:. v_(f)=v_(0)+2nv=v_(0)+(v_(0))/2 =(3v_(0))/2`
`:. k_(f)=1/2 mv^(f)^(2)=1/2m((3v_(0))/2)^(2)=9/4 (1/2 mv_(0)^(2))=9/4 k_(i) ne 4k_(i)`
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