Question

Which of the following statements are correct about a planet revolving around the sun in an elliptical orbit?
A) Its kinetic energy is constant.
B) Its angular momentum is constant.
C) Its areal velocity is constant.
D) Its time period is proportional to ${r^3}$.

Answer 1

Just remember that when a planet revolves around the sun in an elliptical orbit, its kinetic energy continuously changes due to non-uniform speed of the planet and its angular momentum is constant as areal velocity is constant. Also remember, all the three Kepler’s law of planetary motion.

Complete step by step solution:
Let us first talk about KEPLER’S law of planetary motion, which are given by
KEPLER’S FIRST LAW: Kepler’s first law states that every planet moves along an ellipse, with the Sun located at a focus of the ellipse.

KEPLER’S SECOND LAW: Kepler’s second law states that the line joining the sun and a planet sweeps out equal areas in equal times, that is, the area divided by time, called the areal velocity, is constant.

KEPLER’S THIRD LAW OF MOTION: Kepler’s third law states that the square of the period is proportional to the cube of the semi-major axis of the orbit. Therefore, from second law we can say that angular momentum of a planet is constant.

Hence, option (B) is correct.

From the second law, we can say that the real velocity of the planet is constant. So, option (B) is also correct. Now, from the third law, we can say that the time period is proportional to cube of semi-major axis.

Therefore, option (D) is also correct.

After studying Kepler’s law deeply, we can say that when a planet moves around the sun in an elliptical orbit, it changes its rotational kinetic energy.

Therefore, option (A) is not correct.

Note: We can derive Kepler’s third law of motion by using the formula of force of gravitation. Let force of gravitation between the sun and the planet and equate it equal to centripetal force. After equating both the forces, you will get the equation of third law:
${T^{2\,}}\,\propto \,{r^3}$
Here, $T$ is the time period and $r$ is the length of the semi-major axis.