Answer
We can rank them in order of the particle's change in kinetic energy, from highest to lowest:
$c \gt b \gt e \gt d \gt a = f$
Work Step by Step
We can use conservation of energy to find an expression for the change in kinetic energy $\Delta K$:
$K_f+U_f = K_i+U_i$
$K_f-K_i = U_i-U_f$
$\Delta K = V_i~q-V_f~q$
$\Delta K = (V_i-V_f)~q$
We can find the change in kinetic energy in each case.
(a) $\Delta K = (V_i-V_f)~q$
$\Delta K = [100~V-(-50~V)]~(-5\times 10^{-9}~C)$
$\Delta K = -7.5\times 10^{-7}~J$
(b) $\Delta K = (V_i-V_f)~q$
$\Delta K = (-50~V-50~V)~(-5\times 10^{-9}~C)$
$\Delta K = 5.0\times 10^{-7}~J$
(c) $\Delta K = (V_i-V_f)~q$
$\Delta K =(50~V-20~V)~(25\times 10^{-9}~C)$
$\Delta K = 7.5\times 10^{-7}~J$
(d) $\Delta K = (V_i-V_f)~q$
$\Delta K = [400~V-(-100~V)]~(-1\times 10^{-9}~C)$
$\Delta K = -5.0\times 10^{-7}~J$
(e) $\Delta K = (V_i-V_f)~q$
$\Delta K = [-100~V-(-250~V)]~(1\times 10^{-9}~C)$
$\Delta K = 1.5\times 10^{-7}~J$
(f) $\Delta K = (V_i-V_f)~q$
$\Delta K = (100~V-250~V)~(5\times 10^{-9}~C)$
$\Delta K = -7.5\times 10^{-7}~J$
We can rank them in order of the particle's change in kinetic energy, from highest to lowest:
$c \gt b \gt e \gt d \gt a = f$
