Question

A body of mass `m=10^(-2) kg` is moving in a medium and experiences a frictional force `F=-KUPSILON^(2).` Its initial speed is `upsilon_(0)=10ms^(-2)`. If , after `10s`, its energy is `(1)/(8)m upsilon_(0)^(2)`, the value of `k` will be
A. `10^(-4) kg m^(-1)`
B. `10^(-1)kg m^(-1)`
C. `10^(-3)kg m^(-1)`
D. `10^(-3)kg m^(-1)`

Answer 1

Here, `m=0^(-2)kg, upsilon_(0)=10 m//s, t=10s`,
`F=-kupsilon^(2),k=?`
If `upsilon_(t)` is velocity of body after `10s`, then
`(1)/(2)m upsilon_(t)^(2)=(1)/(8)m upsilon_(0)^(2):. upsilon_(t)=(upsilon_(0))/(2)=(10)/(2)=5 m//s`
From `F=-kupsilon^(2)`
`m(d upsilon)/(dt)=-k upsilon^(2)`
`(d upsilon)/(dt)=(-k)/(m)upsilon^(2)=-100k upsilon^(2)`
or `(dupsilon)/(upsilon^(2))=-100k dt`
`int_(10)^(5)(dupsilon)/(upsilon^(2))=-100k int_(0)^(10)dt`
`(1)/(5)-(1)/(10)=100k(10-0)`
`(1)/(10)=100xx10 k dt`
`k=10^(-4)`