How is the kinetic energy of a moving cart affected if (a) its mass is doubled, (b)its velocity
is reduced to `(1)/(3)rd` of the initial velocity ?

Explanation

(a)`K_(1)=(1)/(2)mv^(2).K_(2)=(1)/(2)xx2mxxv^(2)`
`therefore K_(2)=2K_(1)`
Alternative :If mass m is doubled (keeping the speed same),the kinetic energy gets doubled (since kinetic energy is directly proportional to the mass)
i.e., Increase in kinetic energy =`K_(2)-K_(1)=2K_(1)-K_(1)`
`=K_(1)` (initial kinetic energy )
(b)`K_(1)=(1)/(2)mv^(2),k_(2)=(1)/(2)mxx((1)/(3)v)^(2)=(1)/(9)xx(1)/(2)mv^(2)`
`therefore K_(2)=(1)/(9)K_(1)`
ALternative :If velocity v is reduced to `(1)/(3)`rd (keeping the mass same),the kinetic energy reduces to `(1)/(9)th` initial value (since kinetic energy is directly proportional to the square of velocity).
i.e. Decreases in kinetic energy =`K_(1)-K_(2)=K_(1)-(1)/(9)K_(1)`
`=(8)/(9)K_(1)` (i.e. `(8)/(9)th` the initial kinetic energy )