Question

A comet has an aphelion distance of 6.1 *10power 11 metre and perihelion distance of 5.1*10power 11 metre. The mass of the sun is 2.0*10power 30 kg. Calculate the speed of the comet at the perihelion and the aphelion.

Answer 1

From the conditions of the problem, we have

$Q=6.1 \cdot 10^{11} m$

$q=5.1 \cdot 10^{11} m$

$M=2 \cdot 10^{30} kg$

1) Define the semimajor axis of the orbit

$a=\frac{Q+q}{2}=\frac{6.1 \cdot 10^{11} m+5.1 \cdot 10^{11} m}{2}=5.6 \cdot 10^{11} m$

2) Determine the circular velocity of the comet in orbit

$V_a=\sqrt{\frac{G(M+m)}{a}}$

 neglecting a small value $m$

we write

$V_a=\sqrt{\frac{G \cdot M)}{a}}=\sqrt{\frac{6.67 \cdot 10^{-11} \cdot 2 \cdot 10^{30})}{5.6 \cdot 10^{11}}}=1.543 \cdot 10^4 m/s=15.43 km/s$

3) Then the speed of the comet at perihelion

$V_q=V_a \cdot\sqrt{\frac{Q}{q}}=15.43 \cdot\sqrt{\frac{6.1 \cdot 10^{11}}{5.1 \cdot 10^{11}}}=16.88 km/s$

4) the speed of the comet in aphelion.

$V_Q=V_a \cdot\sqrt{\frac{q}{Q}}=15.43 \cdot\sqrt{\frac{5.1 \cdot 10^{11}}{6.1 \cdot 10^{11}}}=14.11 km/s$