Electric Potential of a Uniform Field is:
V = -E*ds, when you take V=0 at point P then the potential at the original position of the charge is -E*s = -E*L*cos(theta), and at the final position you have V = -E*L (because cos(0) = 1).
Now use conservation of energy to solve:
Ki + Ui = Kf + Uf (K=kinetic, U=potential, i=initial, f=final)
plug in the the components.
0 - q*E*Lcos(theta) = (1/2)*m*v^2 - q*E*L
where q = 1.9*10^-6 C (remember to convert from micrometers to meter)
E = 280 V/m
L = 1.6 m
m = 0.010 kg
theta = 60
Now plug in the variables and rearrange the equation solving for v:
v = squareroot ((2*(-q*E*L*cos(theta) + q*E*L) / m))
v = squareroot ((2*(-1.9*10^-6 C*280*1.6*cos(60) + 1.9*10^-6 C*280*1.6 m) / 0.010 kg))
v = 0.292 m/s