A pointlike projectile of mass m is fired upward from theorigin O. At any time after launch, the position and velocity ofthe projectile are described by the variables x,y, x' and y' (wherex' is the time derivative of x and y' the time derivative of y) oralternatively by the vectors r=ix+jy andv=ix'+jy'.

write the vector angular momentum L and the torqueT about the origin in terms of x,y,x',y', i,j.

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A particle of mass m moves under the influence of a central force.The potential energy function is

V(r) = {
Vo if r < a
0 if r > a

as illustrated in the figure. The initial position of the particleis (x,y) = (a/2, 0).

The initial velocity is (vx,vy) = (0, v0). Determine the finalvelocity.
[Data: m = 1 kg; V0 = 10 J; v0 = 4.6 m/s.]

(A) What is the final speed?

(B) What is the final direction Î¸?

**HINT GIVEN**

Energy and angular momentum are conserved.

Express E and L in terms of the initial conditions. Express Eand L in terms of the coordinates when the
particle passes the radius r = a. Set the energies equal, and theangular momenta equal. That yields two
equations for two unknowns. The rest is just math.

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