The potential energy of a particle of mass 5 kg moving in the x-y plane is given by `U=-7x+24y` joule, x and y being in metre. Initially at t = 0 the particle is at the origin. (0, 0) moving with a velocity of `6[2.4hat(i)+0.7hat(j)]m//s`. The magnitude of force on the particle is :

Options

A. The particle has speed `20 m//s` at `t=4` sec
B. The particle has an acceleration `25 m//s^(2)`
C. The acceleration of particle is normal to its intial velocity.
D. None of the above each correct

Answer

C

Explanation

Force acting on the particle
`vec(F)=-(delU)/(delx)hat(i)-(delU)/(dely)hat(j)=7hat(i)-24hat(j)`
Accerleration of particle
`=(F)/(m)=sqrt(7^(2)+24^(2))/(5)=(25)/(5)=5ms^(-2)`
velocity of particle
`vec(v)=vec(u)+vec(a)t=(14.4hat(i)+4.2hat(j))+(1.4hat(i)-4.8hat(j))t`
Speed at `4` sec`=sqrt((14.4+5.6)^(2)+(19.2-4.2)^(2))`
`=25ms^(-1)`
Now `vec(a).vec(u)=(1.4hat(i)-4.8hat(j)).(14.4hat(i)+4.2hat(j))=20.16-20.16=0rArr` The acceleration of particle is normal to its initial velocity.