Question

ENERGY OF ORBITTING SATELLITES
39.A particle fall towards earth from indinity.The velocity with which it reaches earths surface is --
40.A stone is dropped from a height equal to nR, where R is the radius of the earth,from the surface of the earth.The velocity of the stone on reaching the surface of the earth is----

Answer 1

39-
Suppose a particle falls from rest from a point A at a distance d from the centre of the earth. Suppose at time t second, the particle is at a distance x from the centre. At this point the velocity of the particle is v. Then, we have a formula for v:

$$\begin{gathered} {\mathrm{v}}^2=2{\mathrm{gr}}^2[\frac{1}{\mathrm{x}}-\frac{1}{\mathrm{d}}] \\ \mathrm{Now},\mathrm{it}\mathrm{is}\mathrm{given}\mathrm{that}\mathrm{the}\mathrm{particle}\mathrm{falls}\mathrm{from}\mathrm{rest}\mathrm{at}\mathrm{infinity}\mathrm{i}.\mathrm{e}.\mathrm{d}=\infty \\ \mathrm{So}, \\ {\mathrm{v}}^2=2{\mathrm{gr}}^2[\frac{1}{\mathrm{x}}-\frac{1}{\infty }] \\ =2{\mathrm{gr}}^2[\frac{1}{\mathrm{x}}-0] \\ \mathrm{or}{\mathrm{v}}^2=\frac{2{\mathrm{gr}}^2}{\mathrm{x}} \\ \mathrm{On}\mathrm{the}\mathrm{surface}\mathrm{of}\mathrm{the}\mathrm{earth},\mathrm{x}=\mathrm{r} \\ \mathrm{Thus},\mathrm{the}\mathrm{velocity}\mathrm{of}\mathrm{the}\mathrm{particle}\mathrm{on}\mathrm{reaching}\mathrm{the}\mathrm{earth}\text{'}\mathrm{s}\mathrm{surface}\mathrm{is} \\ {\mathrm{v}}^2=2\mathrm{gr} \\ \mathrm{or}\mathrm{v}=\sqrt{2\mathrm{gr}} \end{gathered}$$