Chapter 14, Problem 23A

Kepler tells us that a planet sweeps out equal areas in equal intervals of time. Four such equal-area ’triangles' are shown.

Rank these quantities from greatest to least.

a. average speed during the time interval

b. acceleration during the time interval

To determine

To Rank: Average speed during the time interval from greatest to least.

Answer to Problem 23A

A > B > D > C.

Explanation of Solution

Introduction:

Since the planet sweeps out equal areas in equal amounts of time, the planet has to travel faster along a bigger arc and slower along a smaller arc.

The average speed of the planet is greater when the planet has to sweep a bigger arc.

Therefore, the average speed ranking is A > B > D > C.

To determine

To Rank: The acceleration during the time interval.

Answer to Problem 23A

A > B > D > C

Explanation of Solution

Introduction:

Acceleration is the change in velocity with respect to time.

While following an arc, the planet can change its direction from positive to negative, or vice versa.

The change in velocity for a bigger arc would be greater than for a smaller arc. Thus, the acceleration is the highest for the longest arc.

Hence, the ranking is A > B > D > C.