A small body starts falling onto the Sun from a distance equal to the radius of the Earth's Orbit - Physics - Gravitation

Question

A small body starts falling onto the Sun from a distance
equal to the radius of the Earth's Orbit The initial velocity of
the body is equal to 0 in the heliocentric reference frame Find the
time taken by the body to fall into the sun Ans=64 5
days

Hansraj Rai · Accepted Answer

If the mass of the sun is M and the radius of the earth's orbit be r.

Consider that the body has velocity v when it is at a distance x from the sun.

So, by conservation of mechanical energy,

$\frac{1}{2} m v^{2} + (- \frac{G M m}{x}) = 0 - \frac{G M m}{r} v = - \sqrt{\frac{2 G M}{r} \times \frac{(r - x)}{x}} \frac{d x}{d t} = - \sqrt{\frac{2 G M}{r} \times \frac{(r - x)}{x}} \int_{0}^{t} 1 . ⅆ t = - \sqrt{\frac{r}{2 G M}} \int_{0}^{r} \sqrt{\frac{x}{r - x}} ⅆ x$

substituting,

$x = r \sin^{2} θ a n d s o l v i n g, w e g e t t = \sqrt{\frac{r}{2 G M}} \times (\frac{π r}{2}) o n s o l v i n g w e g e t, t = (\frac{\sqrt{2}}{8}) T$

T is the time period of revolution of the earth.

so, t = 64.5 days.

A small body starts falling onto the Sun from a distance equal to the radius of the Earth's Orbit .The initial velocity of the body is equal to 0 in the heliocentric reference frame .Find the time taken by the body to fall into the sun .Ans=64.5 days