3 Marks Question

Question 13 Marks

What is the direction of areal velocity of the earth around the sun?

Answer

Areal velocity of the earth around the sun is given by

$\frac{\vec{\text{dA}}}{\text{dt}}=\frac{\vec{\text{L}}}{2\text{M}}$

where, L is the angular momentum and M is the mass of the earth.

But angular momentum

$\vec{\text{L}}=\vec{\text{r}}\times\vec{\text{p}}=\vec{\text{r}}\times\text{m}\vec{\text{v}}$

Therefore, the direction of a real velocity$\Big(\frac{\vec{\text{dA}}}{\text{dt}}\Big)$is in the direction of $(\vec{\text{r}}\times\vec{\text{v}}),$ i.e., perpendicular to the plane containing r and v and directed as given by right hand rule. So, areal velocity is normal to the plane containing Earth and Sun as shown in the figure.

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Question 23 Marks

Out of aphelion and perihelion, where is the speed of the earth more and why?

Answer

The earth revolves around the sun in an elliptical orbit or by Kepler’s first law and sun remains at its one focus. The position of earth at P and A at shortest and longest distance are called perihelion and Aphelion respectively.

According to the second law of kepler’s the areal velocity of planet around the sun is constant.

$\frac{\text{dA}}{\text{dt}}=\frac{\text{L}}{2\text{m}}=\frac{\text{r}\times\text{p}}{2\text{m}}=\frac{\text{r}\times\text{mv}}{2\text{m}}=\frac{1}{2}\text{r}\times\text{v}$

Hence, if r increases at Aphelion the v decreases and vice - versa at p.

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Question 33 Marks

Molecules in air in the atmosphere are attracted by gravitational force of the earth. Explain why all of them do not fall into the earth just like an apple falling from a tree.

Answer

The air molecules move randomly as they possess thermal (temperature) energy. Air molecules and apples both are attracted by the earth by gravitational force but the resultant velocity of an air molecule is not exactly downward as in case of an apple (as apple has no other direction of motion except downward).

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Question 43 Marks

Draw areal velocity versus time graph for mars.

Answer

Areal velocity of Mars revolving around the Sun does not change with time according to Kepler’s law, i.e. it is constant with time. Then graph of a real velocity versus time is a straight line parallel to time axis.

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