An asteroid in an elliptical orbit about the sun travels at 3.9 106 m/s at perihelion (the point of closest approach) at a distance of 3.7 108 km from the sun. How fast is it traveling at aphelion (the most distant point), which is 8.6 108 km from the sun?

Well, well, well, it looks like this asteroid is quite the speedster! So, let's calculate its speed at aphelion.

First things first, let's convert those distances from kilometers to meters. The perihelion distance of 3.7 x 10^8 km becomes 3.7 x 10^11 m, and the aphelion distance of 8.6 x 10^8 km becomes 8.6 x 10^11 m.

Now, we need to use a little physics magic here. We know that the angular momentum of an object in an elliptical orbit is conserved. That means the product of its mass (which we'll assume to be constant) and its velocity remains the same at all points along its orbit.

So, at perihelion, the velocity is given as 3.9 x 10^6 m/s, and at aphelion, we need to find the corresponding velocity.

Using the conservation of angular momentum, we can set up the equation:

(mass x velocity at perihelion) = (mass x velocity at aphelion)

Simplifying the equation, we get:

(3.9 x 10^6 m/s) = (mass x velocity at aphelion)

Solving for the velocity at aphelion, we find:

velocity at aphelion = (3.9 x 10^6 m/s) / (mass)

But wait! We don't even know the mass of the asteroid. So, unless we have the asteroid's weight-loss plan or the number for Jenny Craig, we can't solve for the aphelion velocity.

Looks like this speedster will remain a mystery!

To find the speed of the asteroid at aphelion, we can make use of the principle of conservation of angular momentum. According to this principle, the product of the asteroid's mass, its linear velocity, and the perpendicular distance from the sun remains constant throughout its orbit.

First, we need to convert the given distances from kilometers to meters. The perihelion distance is 3.7 x 10^8 km, which is equal to 3.7 x 10^11 m. The aphelion distance is 8.6 x 10^8 km, which is equal to 8.6 x 10^11 m.

We know the speed at perihelion is 3.9 x 10^6 m/s.

Next, we can set up the equation using the conservation of angular momentum:

Mass of the asteroid x Linear velocity at perihelion x Perihelion distance = Mass of the asteroid x Linear velocity at aphelion x Aphelion distance

We can cancel out the mass of the asteroid, as it appears on both sides of the equation. Rearranging the equation, we get:

Linear velocity at aphelion = (Linear velocity at perihelion x Perihelion distance) / Aphelion distance

Plugging in the given values, we have:

Linear velocity at aphelion = (3.9 x 10^6 m/s x 3.7 x 10^11 m) / (8.6 x 10^11 m)

Simplifying the expression, we get:

Linear velocity at aphelion = 1.68 x 10^6 m/s

Therefore, the asteroid is traveling at approximately 1.68 x 10^6 m/s at aphelion.