Answer
The comet's speed at perihelion is 55.1 km/s
Work Step by Step
We can use conservation of energy to find the comet's speed $v_p$ at perihelion:
$\frac{1}{2}mv_p^2-\frac{G~M_s~m}{R_p} = \frac{1}{2}mv_a^2-\frac{G~M_s~m}{R_a}$
$v_p^2 = v_a^2+\frac{2G~M_s}{R_p} -\frac{2G~M_s}{R_a}$
$v_p^2 = v_a^2+2~G~M_s~(\frac{1}{R_p} -\frac{1}{R_a})$
$v_p = \sqrt{v_a^2+2~G~M_s~(\frac{1}{R_p} -\frac{1}{R_a})}$
$v_p = \sqrt{(1.00\times 10^4~m/s)^2+(2)(6.67\times 10^{-11}~m^3/kg~s^2)~(1.989\times 10^{30}~kg)~(\frac{1}{8.9\times 10^{10}~m} -\frac{1}{5.3\times 10^{12}~m})}$
$v_p = 5.51\times 10^4~m/s = 55.1~km/s$
The comet's speed at perihelion is 55.1 km/s.
