Skip to main content

Visit this link for 1 : 1 LIVE Classes.

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

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

Answer: (4)

Popular posts from this blog

${\log _{\sqrt 5 }}\left[ {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right]$

The range of the function $f(x) = {\log _{\sqrt 5 }}\left[ {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right]$ is: (A) $[ - 2,2]$ (B) $\left[ {\frac{1}{{\sqrt 5 }},\sqrt 5 } \right]$ (C) $(0,\sqrt 5 )$ (D) $[ 0,2]$ Solution We have, $f(x) = {\log _{\sqrt 5 }}\left( {3 - 2\sin \frac{{3\pi }}{4}\sin x + 2\cos \frac{\pi }{4}\cos x} \right)$ $ \Rightarrow f(x) = {\log _{\sqrt 5 }}\left[ {3 + \sqrt 2 (\cos x - \sin x)} \right]$ Now, $ - \sqrt 2  \le \cos x - \sin x \le \sqrt 2 $ $\therefore - 2 \le \sqrt 2 (\cos x - \sin x) \le 2$ $\therefore 1 \le 3 + \sqrt 2 (\cos x - \sin x) \le 5$ $\therefore{\log _{\sqrt 5 }}1 \le {\log _{\sqrt 5 }}[3 + \sqrt 2 (\cos x - \sin x)] \le {\log _{\sqrt 5 }}5$ $ \Rightarrow 0 \le f(x) \le 2$ Answer: (D)