Question

A planet follows an elliptical orbit that brings it as close as 46 million km to a star and as far as 85 million km from the star. At both of these locations the velocity of the planet makes a right angle to the direction to the star. If the planet's speed is 38 km/s when it is farthest from the star, how fast is it moving when it is closest to the star?

Answer 1

70.62 km/s.

We can use the conservation of angular momentum to solve this problem.

According to this principle, the product of the planet's mass, its velocity, and its distance from the star is constant throughout its orbit.

Therefore, we can write:

mv1r1 = mv2r2

where
m is the planet's mass,

v1 and v2 are its velocities at the closest and farthest points in its orbit, and

r1 and r2 are its distances from the star at those points.

We are given that r1 = 46 million km and r2 = 85 million km.

We also know that v2 = 38 km/s. To find v1, we can rearrange the above equation as:

v1 = (v2r2r2)/r1/r1

Plugging in the given values, we get:

v1 = (38 km/s * 85 million km * 85 million km) / (46 million km * 46 million km)
v1 = 70.62 km/s

Therefore, the planet's speed when it is closest to the star is approximately 70.62 km/s.

Answer 2

We can use the conservation of angular momentum to solve this problem. According to this principle, the angular momentum of a planet in its elliptical orbit is conserved, meaning that the product of its orbital velocity and its distance from the star is constant.

Let's denote the planet's speed when it is closest to the star by v1, and its speed when it is farthest from the star by v2. Similarly, let's denote the planet's distance from the star when it is closest and farthest by r1 and r2, respectively.

Using the conservation of angular momentum, we can write:

v1 * r1 = v2 * r2

Plugging in the values given in the problem, we get:

v1 * 46 million km = 38 km/s * 85 million km

Solving for v1, we get:

v1 = (38 km/s * 85 million km) / 46 million km

v1 = 70 km/s

Therefore, the planet is moving at a speed of 70 km/s when it is closest to the star.