An asteroid in orbit about the sun has a linear speed of 4 km/s when at a distance of closest approach d from the sun.What is its linear speed when it is at its greatest distance from the sun,a distance 2 d?
A) 1 km/s
B) 2 km/s
C) 8 km/s
D) 16 km/s

Question

Quizplus · Accepted Answer

According to Kepler's second law, the line connecting a planet to the sun sweeps out equal areas in equal intervals of time. This means that when the planet is closest to the sun, it moves faster and when it is farthest, it moves slower.

The conservation of angular momentum can also be applied to the asteroid's orbit:

angular momentum = moment of inertia x angular velocity

Since the moment of inertia of the asteroid is constant, when it is closer to the sun and moving faster, its angular velocity is higher. Therefore, when the asteroid is at its farthest distance, the angular velocity must be lower to maintain the same angular momentum.

Using the equation for angular velocity:

angular velocity = linear velocity / distance from the sun

we can calculate the new linear velocity when the asteroid is at a distance of 2d:

angular velocity at 2d = (4 km/s) / d

angular velocity at d = (v2) / (2d)

Setting the two equations equal and solving for v2, we get:

(4 km/s) / d = (v2) / (2d)

v2 = 8 km/s

Therefore, the asteroid's linear speed when it is at its greatest distance from the sun is 2 km/s.

An Asteroid in Orbit About the Sun Has a Linear