Verified Solution • 2025–26 Session moderate

QJEEM22Ph11eb: JEE Main 2022 Physics - Class 11 JEEM22Ph11eb Solution: An object is projected in the air with initial velocity u at an …

An object is projected in the air with initial velocity u at an angle θ. The projectile motion is such that the horizontal range R, is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be _________ degree.

Step-by-Step Verified Solution

<mjx−containerclass="MathJaxCtxtMenuAttached0"jax="CHTML"display="true"tabindex="0"ctxtmenucounter="1"style="font−size:130.9<mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="1" style="font-size: 130.9%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 12.11em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-bottom: 0.15em;"><mjx-msub><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c78"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.404em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c></mjx-mi></mjx-texatom></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-texatom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mi class="mjx-n"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><mjx-container class="MathJax CtxtMenu_Attached_0" jax="CHTML" display="true" tabindex="0" ctxtmenu_counter="3" style="font-size: 129.3%; position: relative;"><mjx-math display="true" class="MJX-TEX" aria-hidden="true" style="margin-left: 0px; margin-right: 0px;"><mjx-mtable style="min-width: 12.11em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-bottom: 0.15em;"><mjx-msub><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: -0.15em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c6D"></mjx-c><mjx-c class="mjx-c61"></mjx-c><mjx-c class="mjx-c78"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-texatom space="2" texclass="INNER"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.404em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-texatom></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mfrac><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c52"></mjx-c></mjx-mi></mjx-texatom></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-texatom texclass="ORD"><mjx-mtext class="mjx-n"><mjx-c class="mjx-cA0"></mjx-c></mjx-mtext><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mrow><mjx-msup><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c75"></mjx-c></mjx-mi></mjx-texatom><mjx-script style="vertical-align: 0.363em;"><mjx-mn class="mjx-n" size="s"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-script></mjx-msup><mjx-mi class="mjx-n" space="2"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi></mjx-mrow></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-texatom texclass="ORD"><mjx-mi class="mjx-n"><mjx-c class="mjx-c67"></mjx-c></mjx-mi></mjx-texatom></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mi class="mjx-n"><mjx-c class="mjx-c73"></mjx-c><mjx-c class="mjx-c69"></mjx-c><mjx-c class="mjx-c6E"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2061"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em; padding-bottom: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd extra="true"></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align: right; padding-right: 0px; padding-top: 0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align: left; padding-left: 0px; padding-top: 0.15em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D703 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msup space="2"><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-script style="vertical-align: 0.413em;"><mjx-texatom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2218"></mjx-c></mjx-mo></mjx-texatom></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><msub><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">R</mi></mrow><mrow data-mjx-texclass="ORD"><mo data-mjx-texclass="OP" movablelimits="true">max</mo></mrow></msub><mo>=</mo><mfrac><mrow><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">u</mi></mrow><mn>2</mn></msup><mi>sin</mi><mo data-mjx-texclass="NONE">⁡</mo><mn>2</mn><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mn>45</mn><mrow data-mjx-texclass="ORD"><mo>∘</mo></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">g</mi></mrow></mfrac><mo>=</mo><mfrac><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">u</mi></mrow><mn>2</mn></msup><mrow data-mjx-texclass="ORD"><mtext>&nbsp;</mtext><mi mathvariant="normal">g</mi></mrow></mfrac></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mfrac><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">R</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">u</mi></mrow><mn>2</mn></msup><mrow><mn>2</mn><mrow data-mjx-texclass="ORD"><mtext>&nbsp;</mtext><mi mathvariant="normal">g</mi></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">u</mi></mrow><mn>2</mn></msup><mi>sin</mi><mo data-mjx-texclass="NONE">⁡</mo><mn>2</mn><mi>θ</mi></mrow><mrow data-mjx-texclass="ORD"><mi mathvariant="normal">g</mi></mrow></mfrac></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi>sin</mi><mo data-mjx-texclass="NONE">⁡</mo><mn>2</mn><mi>θ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mn>2</mn><mi>θ</mi><mo>=</mo><msup><mn>30</mn><mrow data-mjx-texclass="ORD"><mo>∘</mo></mrow></msup><mo>,</mo><msup><mn>150</mn><mrow data-mjx-texclass="ORD"><mo>∘</mo></mrow></msup></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd></mtd><mtd><mi>θ</mi><mo>=</mo><msup><mn>15</mn><mrow data-mjx-texclass="ORD"><mo>∘</mo></mrow></msup><mo>,</mo><msup><mn>75</mn><mrow data-mjx-texclass="ORD"><mo>∘</mo></mrow></msup></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container></mjx-assistive-mml></mjx-container>

Related Learning Resources & Exam Preparation