Given:
\[ V = \frac{2\pi R}{T} \]
The maximum height attained by the particle is given by:
\[ H = \frac{v^2 \sin^2 \theta}{2g} \]
We are given that:
\[ 4R = \frac{4\pi^2 R^2 \sin^2 \theta}{T^2 \cdot 2g} \]
Simplifying:
\[ \sin^2 \theta = \frac{2gT^2}{\pi^2 R} \]
Taking the square root:
\[ \sin \theta = \sqrt{\frac{2gT^2}{\pi^2 R}} \]
Thus:
\[ \theta = \sin^{-1} \left( \sqrt{\frac{2gT^2}{\pi^2 R}} \right) \]
A particle is projected at an angle of \( 30^\circ \) from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is \( h_0 \) and height traversed in the last second, before it reaches the maximum height, is \( h_1 \). The ratio \( \frac{h_0}{h_1} \) is __________. [Take \( g = 10 \, \text{m/s}^2 \)]
Let \( y = f(x) \) be the solution of the differential equation
\[ \frac{dy}{dx} + 3y \tan^2 x + 3y = \sec^2 x \]
such that \( f(0) = \frac{e^3}{3} + 1 \), then \( f\left( \frac{\pi}{4} \right) \) is equal to:
Find the IUPAC name of the compound.
If \( \lim_{x \to 0} \left( \frac{\tan x}{x} \right)^{\frac{1}{x^2}} = p \), then \( 96 \ln p \) is: 32