A-ball-is-projected-from-a-point-O-with-an-initial-velocity-u-and-angle-with-the-horizontal-ground-Given-that-it-travels-such-that-it-just-clears-two-walls-of-height-h-and-distances-2h-and-4h-fro

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

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

Answered by mr W last updated on 17/Apr/20

$$sincethetrackoftheballisaparabola,$$$$youcangettheanswerwithoutmuch$$$$calcuation:$$$$\left(c\right)duetosymmetry,therangeis$$$$L=2h+2h+2h=6h$$$$$$$$themaximumheightoftheball{h}_{max}$$$$\frac{h_{max}}{h_{max}-h}={\left(\frac{3h}{h}\right)}^2=9$$$$\Rightarrow h_{max}=\frac{9}{8}h$$$$$$$$\left(a\right)$$$$\tan \theta =\frac{2h_{max}}{3h}=\frac{3}{4}$$$$$$$$\left(b\right)$$$$totalflighttime=T$$$$h_{max}=\frac{1}{2}g{\left(\frac{T}{2}\right)}^2=\frac{9h}{8}$$$$\Rightarrow T=3\sqrt{\frac{h}{2g}}$$

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

$$sorrysirwhere3hcamefrom?$$

Commented by mr W last updated on 17/Apr/20

$$horizontaldistanceoftheballtoO$$$$is3hwhenitisatitshighestposition.$$

Commented by mr W last updated on 17/Apr/20

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

$$thankyounowitsclear.$$

Commented by Rio Michael last updated on 17/Apr/20

$$\mathrm{sir}\mathrm{why}\mathrm{would}\mathrm{you}\mathrm{say}\mathrm{it}\mathrm{travels}\mathrm{a}\mathrm{distance}\mathrm{of}2h\mathrm{after}\mathrm{the}\mathrm{second}\mathrm{wall}?.$$

Commented by mr W last updated on 17/Apr/20

$$atx=2htheballhasaheighth$$$$atx=4htheballhasaheighth$$$$\Rightarrow inthemiddle,i.e.atx=3h,the$$$$ballreaches{h}_{max},becausetheparabola$$$$issymmetricrightorlefttothe$$$$highestpoint.youcangettherest.$$

Commented by Rio Michael last updated on 17/Apr/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

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

$$whyyousquare{\left(\frac{3h}{h}\right)}^2?$$

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

$$whereissquarecame$$$$from?$$

Commented by mr W last updated on 21/Apr/20

$$thetrackofballisaparabola!$$$$withaparabolae.g.y={ax}^{2}wehave$$$$y_1={ax}_1^2$$$$y_2={ax}_2^2$$$$\frac{y_1}{y_2}={\left(\frac{x_1}{x_2}\right)}^2$$

Commented by peter frank last updated on 21/Apr/20

$$thankyou$$

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