Arman G. answered • 12/08/19
Aerospace Engineer
The force is conservative because you can define a potential function P such that F = grad(P)... This is trivial as P could just be P = 3x + 5y...
a) Work = ∫C F · ds = (by fundamental theorem of line integrals) = P(b) - P(a) =P(2,-3) - P(0,0) = -9 joules... This is path independent because of P, if P didn't exist the path would be important but since a potential function does exist it's a conservative vector field and is path independent because the integral just relies on the endpoints.
b) work = change in kinetic energy, 9 = 0.5*m*v2 ... solve for v and if we assume it's in the direction of the final position so you can multiply it times the unit vector in that direction which is (2/sqrt(13))*i, (-3/sqrt(13))*j... v is not necessarily in this final direction though... and energy is a scalar so I'm not sure how you would find V vector for sure... just the magnitude... but I guess you are assuming it's moving in a straight path for this problem so it would be in that direction... but note that work is independent of path so if it took a non-straight path to get there it would still have the same magnitude of velocity, but it might be in a different direction.
c) Potential energy change is equal to kinetic energy change but negative, if potential increases by 9 then kinetic decreases by 9... so total energy is constant, that's why it's called a conservative vector field! Work is change in kinetic energy which just is the change in potential energy... this is because potential energy function is negative of potential function so that the sum of the potential and kinetic is constant.
