Question

The drawing below is NOT drawn to scale
The drawing represents a planet orbiting some star. The planet is close to the star as it moves from point $1$ to point $2$ than it is when it moves from point $3$ to point $4$ in its orbit. The dotted line shows the orbital path.
If the time it takes the planet to get from point $1$ to point $2$ is equal to the time it takes the planet to get from point $3$ to point $4$, then which of the following statements is true, according to Kepler's laws of planetary motion?

• A. The average speed as the planet travels from $1$ to $2$ is the same as when it travels from $3$ to $4$
• B. Area $1$ in the diagram is equal to Area $2$
• C. Area $1$ in the diagram is smaller than Area $2$
• D. The planet moves further from $3$ to $4$ than from $1$ to $2$
• E. The average speed as the planet moves from $3$ to $4$ is greater than when the planet moves from $1$ to $2$

Answer 1

Answer: (B)

Area $1$ in the diagram is equal to Area $2$

Kepler's second law states that areal velocity of a planet remains constant i.e. a planet sweeps equal area in equal time .

Areal velocity of planet from 1 to 2 ,

$v_A=A_1/\Delta t$ ,

Areal velocity of planet from 3 to 4 ,

$v_A^{\prime }=A_2/\Delta t$ ,

given that time taken by planet to cover area 1 and area 2 is same ,

but by Kepler's law $v_A=v_A^{\prime }$ ,

or $A_1/\Delta t=A_2/\Delta t$ ,

or $A_1=A_2$