Question

Figure (11-Q2) shows the elliptical path of a planet about the sun. The two shaded parts have equal area. If t1 and t2 be the time taken by the planet to go from a to b and from c to d respectively,
(a) t1 < t2
(b) t1 = t2
(c) t1 > t2
(d) insufficient information to deduce the relation between t1 and t2.
Figure

Answer 1

(c) If t1 and t2 is the time taken by the planet, then t1 > t2.

Explanation:

Step 1:

As per the Kepler’s 2nd law, in the orbit, the speed varies in such a way that the vector radius joining sun with planet sweet out same areas in equal time intervals.

It is also to be understood that in equal intervals of time, the planet sweeps equal.  

Step 2:

Kepler's 2nd law  

Area of velocity =$\mathrm{d} \mathrm{A} / \mathrm{dt}$

dA/dt = $=1(r)(V d t) / 2 d t=1 / 2 r V$

$\mathrm{d} \mathrm{A} / \mathrm{dt} \rightarrow$Areal velocity

$d A \rightarrow$ small area traced

wherein

Step 3:

Similar to Law of conservation of momentum

$\mathrm{d} \mathrm{A} / \mathrm{dt}=\mathrm{L} / 2 \mathrm{m}$

l--> Angular momentum

Known as Law of Area, planet’s areal velocity is constant in elliptical path. This implies equal area will be covered in equal time.

$t 1=t 2$