Question Number 29400 by ajfour last updated on 08/Feb/18
Commented by ajfour last updated on 08/Feb/18
$${Find}\:\boldsymbol{{b}}\:{in}\:{terms}\:{of}\:\boldsymbol{{a}},\boldsymbol{{u}},\boldsymbol{\alpha},\:{and}\:\boldsymbol{{e}}\:. \\ $$$$\left(\boldsymbol{{e}}\:{being}\:{the}\:{coefficient}\:{of}\right. \\ $$$$\left.{restitution}\right)\: \\ $$
Answered by 33 last updated on 08/Feb/18
$${a}\:=\:\left({u}\:{cos}\:\alpha\:\right)\:{t}_{\mathrm{1}} \\ $$$${b}\:=\:\left({eu}\:{cos}\:\alpha\:\right)\:{t}_{\mathrm{2}} \\ $$$${t}_{\mathrm{1}\:} +\:{t}_{\mathrm{2}\:\:} =\:{t} \\ $$$$\frac{{a}}{{u}\:{cos}\:\alpha\:}\:+\:\frac{{b}}{{eu}\:{cos}\:\alpha\:}\:=\:\frac{\mathrm{2}{u}\:{sin}\:\alpha}{{g}} \\ $$$$\Rightarrow\:{b}\:=\:\left(\:\:\frac{{e}\:{u}^{\mathrm{2}} \:{sin}\:\mathrm{2}\alpha}{{g}}\:−\:{ae}\:\right) \\ $$
Commented by ajfour last updated on 08/Feb/18
$${Thanks}. \\ $$
Answered by mrW2 last updated on 08/Feb/18
Commented by mrW2 last updated on 08/Feb/18
$${L}=\frac{{u}^{\mathrm{2}} \:\mathrm{sin}\:\mathrm{2}\alpha}{{g}} \\ $$$${b}'={L}−{a} \\ $$$${b}={eb}'={e}\left({L}−{a}\right)={e}\left(\frac{{u}^{\mathrm{2}} \:\mathrm{sin}\:\mathrm{2}\alpha}{{g}}−{a}\right) \\ $$$${the}\:{wall}\:{acts}\:{like}\:{a}\:{mirror}. \\ $$
Commented by 33 last updated on 09/Feb/18
$${but}\:{sir}\:{how}\:{can}\:{we}\:{use}\:{the}\:{eq}\:{of} \\ $$$${projectile}\:{motion}\:{when}\:{we}\:{know} \\ $$$${that}\:{the}\:{path}\:{from}\:{A}\:{to}\:{B}\:{is}\:{not} \\ $$$${at}\:{all}\:{parabolic}. \\ $$$${Sir},\:{i}\:{do}\:{not}\:{think}\:{it}\:{can}\:{be}\:{done} \\ $$$${in}\:{this}\:{way}\:{tho}\:{you}\:{got}\:{the}\:{same} \\ $$$${snswer}.\left(\:{no}\:{offence}\right) \\ $$
Commented by mrW2 last updated on 09/Feb/18
$${To}\:{get}\:{the}\:{result}\:{that}\:{b}={eb}',\:{we}\:{even} \\ $$$${don}'{t}\:{need}\:{to}\:{know}\:{any}\:{equation}. \\ $$$$ \\ $$$${Assume}\:{the}\:{ball}\:{hits}\:{the}\:{wall}\:{at} \\ $$$${point}\:{D}. \\ $$$${Let}'{s}\:{image}\:{that}\:{the}\:{ball}\:{becomes}\:{two} \\ $$$${balls}\:{after}\:{the}\:{hit}.\:{One}\:{ball}\:{continues}\:{the} \\ $$$${original}\:{motion}\:{to}\:{the}\:{right}\:{as}\:{if}\:{the} \\ $$$${wall}\:{doesn}'{t}\:{exist}.\:{The}\:{other}\:{ball}\:{is} \\ $$$${reflected}\:{by}\:{the}\:{wall}\:{and}\:{goes}\:{to}\:{the} \\ $$$${left}.\:{Both}\:{balls}\:{have}\:{the}\:{same}\: \\ $$$${vertical}\:{speed}\:{at}\:{point}\:{D},\:{but}\:{different}\: \\ $$$${horizontal}\:{speed}.\:{Since}\:{they}\:{have} \\ $$$${the}\:{same}\:{vertical}\:{start}\:{speed},\:{they} \\ $$$${need}\:{the}\:{same}\:{time}\:{to}\:{hit}\:{the}\:{ground}. \\ $$$${And}\:{the}\:{horizontal}\:{distance}\:{each}\:{ball}\: \\ $$$${covers}\:{in}\:{this}\:{time}\:{is}\:{proportional} \\ $$$${to}\:{its}\:{horizontal}\:{speed}\:{at}\:{point}\:{D}. \\ $$$${The}\:{ball}\:{to}\:{the}\:{right}\:{follows}\:{the}\:{path} \\ $$$${DB}\:{with}\:{horizontal}\:{distance}\:{b}'\:{and} \\ $$$${the}\:{ball}\:{to}\:{the}\:{left}\:{follows}\:{the}\:{path} \\ $$$${DC}\:{with}\:{horizontal}\:{distance}\:{b}.\:{Since} \\ $$$${the}\:{horizontal}\:{speed}\:{of}\:{the}\:{ball}\:{to}\:{the} \\ $$$${left}\:{is}\:{e}\:{times}\:{of}\:{the}\:{horizontal}\:{speed} \\ $$$${of}\:{the}\:{ball}\:{to}\:{the}\:{right},\:{therefore} \\ $$$${b}={eb}'. \\ $$$${So}\:{I}\:{think}\:{my}\:{method}\:{is}\:{correct} \\ $$$${and}\:{its}\:{result}\:{is}\:{correct}\:{not}\:{just}\:{by} \\ $$$${accident}. \\ $$$${Please}\:{think}\:{about}\:{it}\:{again}\:{sir}. \\ $$
Commented by 33 last updated on 10/Feb/18
$${yes}\:{sir}\:{you}\:{are}\:{right}.\:{it}\:{be}\:{done} \\ $$$${that}\:{way}.\:{rather}\:{its}\:{better}\:{method} \\ $$$${and}\:{encourages}\:{higher}\:{order} \\ $$$${thinking}.\:{Excellent}\:! \\ $$
Commented by mrW2 last updated on 10/Feb/18
$${Thank}\:{you}\:{for}\:{checking}! \\ $$
Commented by 33 last updated on 10/Feb/18
$${no}\:{problem}\:{sir} \\ $$