The perihelion of the comet totas is 1.69 au and the aphelion is 4.40 au. given that its speed at perihelion is 28 km/s, what is the speed (in km/s) at aphelion (1 au = 1.496 ✕ 1011 m)? (hint: you may use either conservation of energy or angular momentum, but the latter is much easier.)

The speed at the aphelion is 10.75 km/s.

Explanation:

The angular momentum is defined as:

(1)

Since there is no torque acting on the system, it can be expressed in the following way:

  (2)

Replacing equation 1 in equation 2 it is gotten:

(3)

Where m is the mass of the comet, is the orbital radius at the aphelion, is the speed at the aphelion, is the orbital radius at the perihelion and is the speed at the perihelion.          

From equation 3 v_{a} will be isolated:    

  (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:

⇒

⇒

⇒

⇒  

Then, finally equation 4 can be used:

Hence, the speed at the aphelion is 10.75 km/s.