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### A particle is released from height H above the ground ....

A particle is released from height H above the ground. At a certain instant its kinetic energy is three times its potential energy. The height h above the ground and the speed of the particle v at that instant are respectively:

(1) $\frac {H}{4},\frac {3gH}{2}$
(2) $\frac {H}{4},\frac {\sqrt {3gH}}{2}$
(3) $\frac {H}{2},\frac {\sqrt {3gH}}{2}$
(4) $\frac {H}{4},\sqrt {\frac {3gH}{2}}$

Solution

Using conservation of mechanical energy we have,

$mgH=KE+PE=4PE=4mgh$ (given, KE = 3PE)

$\Rightarrow h=\frac {H}{4}$

Now, $\frac {1}{2}mv^2=3mgh=3mg \frac {H}{4}$

$\Rightarrow v=\sqrt \frac {3gH}{2}$

### Sum of the coefficients in the expansion of $(x+y)^n$ ....
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$