Question #104268

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.

Expert's answer

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"

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