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Q.
The equation of projectile is $y = ax - bx^2$ where $a$ and $b$ are the constants. Its horizontal range is
UP CPMTUP CPMT 2011Motion in a Plane
Solution:
The equation of the trajectory of a projectile motion is
$y = x \,tan \,\theta -\frac{gx^{2}}{2u^{2} cos^{2} \theta}$
$ y = x\, tan\, \theta \left(1- \frac{gx}{u^{2}2 \,tan \,\theta cos^{2}\theta}\right)$
$= x \,tan \,\theta\left(1-\frac{gx}{u^{2} sin \,2\theta}\right) $
$= x\, tan \,\theta\left(1- \frac{x}{R}\right)\quad...\left(i\right) $
where $R$ is the horizontal range of the projectile From the given equation of projectile
$ y = ax -bx^{2} $
$ = ax \left(1-\frac{bx}{a}\right)\quad...\left(ii\right)$
Comparing $\left(i\right)$ and $\left(ii\right)$, we get
$ \frac{1}{R} = \frac{b}{a}$ or $R = \frac{a}{b}$