Question

Show that the projection angle θ₀ for a projectile launched from the origin is given by $\theta_0 = tan^-1 [\frac{4hm}{R}]$ where the symbols have their usual meaning

Answer 1

we know, from formula of maximum vertical height , $h_m=\frac{u^2sin^2\theta}{2g}$

Where , u is initial velocity of projectile, $\theta$ is angle made by projectile and g is acceleration due to gravity.
so, $h_m=\frac{u^2sin^2\theta_0}{2g}$....
(1)

formula of horizontal range, $R=\frac{u^2sin2\theta}{g}$
so,Horizontal range, $R=\frac{u^2sin2\theta_0}{g}$ .....… (ii)

From equations (1) and (2),
$\frac{h_m}{R}=\frac{u^2sin^2\theta_0}{2g}/\frac{u^2sin2\theta_0}{g}\\\\\frac{h_m}{R}=\frac{sin\theta_0}{4cos\theta_0}\\\\\frac{4h_m}{R}=tan\theta_0\\\\\theta_0=tan^{-1}\left(\frac{4h_m}{R}\right)$

Answer 2

A tungsten wire of length 20 cm is stretched by 0.1 cm. Find the strain on the wire.The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For large angles of oscillation, a more involved analysis shows that T is greater than $2\pi \sqrt{\frac {I}{g}}$. Think of a quantitative argument to appreciate this resultWhich of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic and (c) non periodic motion? Give period for each case of periodic motion (ω is any positive constant) a. sign ωt - cos ωt b. sin² ωt c. 3 Cos (π/4 - ωt) d. cos ωt + cos 3ωt + cos 5ωt e. exp(-ω²t²) f. 1 + ωt + ω²t²