Suppose earth's orbital motion around the sun is suddenly stopped. Find the time (in days) taken by the earth to fall into the sun.

When the earth's motion is suddenly stopped, it would fall into the sun and (suppose) it comes back. If the effect of temperature of sun is ignored, we can say that the earth would continue to move along a strongly extended flat ellipse whose extreme points are located at the earth's orbit and at the center of the sun.

The semi-major axis of such an ellipse is $\frac{\mathrm{R}}{2}.$

$\frac{{\mathrm{T}}^{\mathrm{\prime }2}}{{\mathrm{T}}^2}=\frac{(\mathrm{R}/2{)}^3}{{\mathrm{R}}^3}$, where T is the time period of normal orbit of the earth.

$\therefore {\mathrm{T}}^{\mathrm{\prime }2}=\frac{{\mathrm{T}}^2}{8}\Rightarrow {\mathrm{T}}^{\mathrm{\prime }}=\frac{\mathrm{T}}{2\sqrt{2}}$

Now, required time is $\mathrm{t}=\frac{{\mathrm{T}}^{\mathrm{\prime }}}{2}=\frac{\mathrm{T}}{4\sqrt{2}}=\frac{365}{4\sqrt{2}}=65\text{ days }$