A ball is projected from a point O with an initial velocity u and angle θ with the horizontal ground. Given that it travels such that it just clears two walls of height h and distances 2h and 4h from O respectively. (a) find the tangent of the angle θ (b) The time of flight of the ball (c) The range of the ball.


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Question Number 89319 by Rio Michael last updated on 16/Apr/20

$A ball is projected from a point O with an initial$ $velocity u and angle θ with the horizontal ground .$ $Given that it travels such that it just clears two walls$ $of height h and distances 2 h and 4 h from O respectively .$ $(a) find the tangent of the angle θ$ $(b) The time of flight of the ball$ $(c) The range of the ball .$
	Answered by mr W last updated on 17/Apr/20

	$s i n c e t h e t r a c k o f t h e b a l l i s a p a r a b o l a,$ $y o u c a n g e t t h e a n s w e r w i t h o u t m u c h$ $c a l c u a t i o n :$ $(c) d u e t o s y m m e t r y, t h e r a n g e i s$ $L = 2 h + 2 h + 2 h = 6 h$ $t h e m a x i m u m h e i g h t o f t h e b a l l h_{m a x}$ $\frac{h_{m a x}}{h_{m a x} - h} = {(\frac{3 h}{h})}^{2} = 9$ $\Rightarrow h_{m a x} = \frac{9}{8} h$ $(a)$ $\tan θ = \frac{2 h_{m a x}}{3 h} = \frac{3}{4}$ $(b)$ $t o t a l f l i g h t t i m e = T$ $h_{m a x} = \frac{1}{2} g {(\frac{T}{2})}^{2} = \frac{9 h}{8}$ $\Rightarrow T = 3 \sqrt{\frac{h}{2 g}}$
	Commented by peter frank last updated on 17/Apr/20

	$s o r r y s i r w h e r e 3 h c a m e f r o m ?$
	Commented by mr W last updated on 17/Apr/20

	$h o r i z o n t a l d i s t a n c e o f t h e b a l l t o O$ $i s 3 h w h e n i t i s a t i t s h i g h e s t p o s i t i o n .$
	Commented by mr W last updated on 17/Apr/20

	Commented by peter frank last updated on 17/Apr/20

	$t h a n k y o u n o w i t s c l e a r .$
	Commented by Rio Michael last updated on 17/Apr/20

	$sir why would you say it travels a distance of 2 h after the second wall ? .$
	Commented by mr W last updated on 17/Apr/20

	$a t x = 2 h t h e b a l l h a s a h e i g h t h$ $a t x = 4 h t h e b a l l h a s a h e i g h t h$ $\Rightarrow i n t h e m i d d l e, i . e . a t x = 3 h, t h e$ $b a l l r e a c h e s h_{m a x}, b e c a u s e t h e p a r a b o l a$ $i s s y m m e t r i c r i g h t o r l e f t t o t h e$ $h i g h e s t p o i n t . y o u c a n g e t t h e r e s t .$
	Commented by Rio Michael last updated on 17/Apr/20

	$thank you sir$
	Commented by peter frank last updated on 20/Apr/20

	$w h y y o u s q u a r e {(\frac{3 h}{h})}^{2} ?$
	Commented by peter frank last updated on 20/Apr/20

	$w h e r e i s s q u a r e c a m e$ $f r o m ?$
	Commented by mr W last updated on 21/Apr/20

	$t h e t r a c k o f b a l l i s a p a r a b o l a!$ $w i t h a p a r a b o l a e . g . y = {a x}^{2} w e h a v e$ $y_{1} = {a x}_{1}^{2}$ $y_{2} = {a x}_{2}^{2}$ $\frac{y_{1}}{y_{2}} = {(\frac{x_{1}}{x_{2}})}^{2}$
	Commented by peter frank last updated on 21/Apr/20

	$t h a n k y o u$
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