Four objects P, Q, R, and S are projected wit | vedclass

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(A) The horizontal range $R$ of a projectile is given by the formula: $R = \frac{u^2 \sin(2	heta)}{g}$.Since the initial velocity $u$ and acceleratio

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$P$

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(A) The horizontal range $R$ of a projectile is given by the formula: $R = \frac{u^2 \sin(2	heta)}{g}$.Since the initial velocity $u$ and acceleration due to gravity $g$ are constant for all objects,the range $R$ depends only on $\sin(2	heta)$.For the given angles:For $P$ $(	heta = 15^o)$: $2	heta = 30^o$,$\sin(30^o) = 0.5$For $Q$ $(	heta = 30^o)$: $2	heta = 60^o$,$\sin(60^o) \approx 0.866$For $R$ $(	heta = 45^o)$: $2	heta = 90^o$,$\sin(90^o) = 1.0$For $S$ $(	heta = 60^o)$: $2	heta = 120^o$,$\sin(120^o) = \sin(60^o) \approx 0.866$Comparing the values,$\sin(2	heta)$ is minimum for $	heta = 15^o$ (which is $P$) and $	heta = 75^o$ (not given). Among the choices,$P$ has the smallest value of $\sin(2	heta)$,therefore $P$ has the minimum range.

Four objects $P, Q, R,$ and $S$ are projected with the same velocity at angles of $15^o, 30^o, 45^o,$ and $60^o$ respectively. Which of these will have the minimum range?

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