The orbit of Halley's Comet around the Sun is a long thin ellipse. At its aphelion (point farthest from the Sun), the comet is 5.1 1012 m from the Sun and moves with a speed of 12.0 km/s. What is the comet's speed at its perihelion (closest approach to the Sun) where its distance from the Sun is 8.2 1010 m?

v=sqrt (GMs/r)

use the first set of data to find the constant sqrt (GMs)= r^2 v

vn =sqrt ((ro^2*vo)/rn)

To find the comet's speed at its perihelion, we can use the principle of conservation of angular momentum. According to this principle, the product of an object's moment of inertia and its angular velocity remains constant as long as no external torques act on the object. In simpler terms, if the object's distance from the rotation axis changes, its rotational speed will also change to maintain the constant product.

In the case of Halley's Comet, since its orbit is an ellipse, its distance from the Sun changes as it moves from the aphelion to the perihelion. Therefore, we can use the conservation of angular momentum to solve for the comet's speed at the perihelion.

The formula for angular momentum is given by:

L = Iω

Where:
L is the angular momentum
I is the moment of inertia
ω is the angular velocity

Since the comet is moving in an elliptical orbit around the Sun, we can relate its angular velocity at the aphelion and perihelion using their respective distances:

r1ω1 = r2ω2

Where:
r1 is the distance of the aphelion (5.1 * 10^12 m)
r2 is the distance of the perihelion (8.2 * 10^10 m)
ω1 is the angular velocity at the aphelion (which we don't know yet)
ω2 is the angular velocity at the perihelion (which we want to find)

Now, let's solve for ω2:

r1ω1 = r2ω2

ω2 = (r1ω1) / r2

ω2 = (5.1 * 10^12 m * 12.0 km/s) / (8.2 * 10^10 m)

Note: We have to convert the speed from km/s to m/s by multiplying it by 1000.

ω2 = (5.1 * 10^12 m * 12.0 km/s * 1000) / (8.2 * 10^10 m)

Calculating this expression gives us:
ω2 ≈ 610.97 m/s

So, the comet's speed at its perihelion is approximately 610.97 m/s.