Answer :
Answer:
The speed at the aphelion is 10.75 km/s.
Explanation:
The angular momentum is defined as:
[tex]L = mrv[/tex] (1)
Since there is no torque acting on the system, it can be expressed in the following way:
[tex]t = \frac{\Delta L}{\Delta t}[/tex]
[tex]t \Delta t = \Delta L[/tex]
[tex]\Delta L = 0[/tex]
[tex]L_{a} - L_{p} = 0[/tex]
[tex]L_{a} = L_{p}[/tex] (2)
Replacing equation 1 in equation 2 it is gotten:
[tex]mr_{a}v_{a} =mr_{p}v_{p}[/tex] (3)
Where m is the mass of the comet, [tex]r_{a}[/tex] is the orbital radius at the aphelion, [tex]v_{a}[/tex] is the speed at the aphelion, [tex]r_{p}[/tex] is the orbital radius at the perihelion and [tex]v_{p}[/tex] is the speed at the perihelion.
From equation 3 v_{a} will be isolated:
[tex]v_{a} = \frac{mr_{p}v_{p}}{mr_{a}}[/tex]
[tex]v_{a} = \frac{r_{p}v_{p}}{r_{a}}[/tex] (4)
Before replacing all the values in equation 4 it is necessary to express the orbital radius for the perihelion and the aphelion from AU (astronomical units) to meters, and then from meters to kilometers:
[tex]r_{p} = 1.69 AU x \frac{1.496x10^{11} m}{1 AU}[/tex] ⇒ [tex]2.528x10^{11} m[/tex]
[tex]r_{p} = 2.528x10^{11} m x \frac{1km}{1000m}[/tex] ⇒ [tex]252800000 km[/tex]
[tex]r_{a} = 4.40 AU x \frac{1.496x10^{11} m}{1 AU}[/tex] ⇒ [tex]6.582x10^{11} m[/tex]
[tex]r_{p} = 6.582x10^{11} m x \frac{1km}{1000m}[/tex] ⇒ [tex]658200000 km[/tex]
Then, finally equation 4 can be used:
[tex]v_{a} = \frac{(252800000 km)(28 km/s)}{(658200000 km)}[/tex]
[tex]v_{a} = 10.75 km/s[/tex]
Hence, the speed at the aphelion is 10.75 km/s.