Menu Close

Author: Tinku Tara

Solve-A-particle-moves-along-the-space-curve-r-t-2-t-i-3t-2-j-2t-3-4t-2-k-find-a-velocity-b-speed-or-magnitude-of-velocity-c-acceleration-d-magnitude-of-acceleration-at-time-t-2-

Question Number 200736 by Calculusboy last updated on 22/Nov/23 $$\boldsymbol{{Solve}}:\:\boldsymbol{{A}}\:\boldsymbol{{particle}}\:\boldsymbol{{moves}}\:\boldsymbol{{along}}\:\boldsymbol{{the}}\:\boldsymbol{{space}} \\ $$$$\boldsymbol{{curve}}\:\underset{−} {\boldsymbol{{r}}}=\left(\boldsymbol{{t}}^{\mathrm{2}} +\boldsymbol{{t}}\right)\boldsymbol{{i}}+\left(\mathrm{3}\boldsymbol{{t}}−\mathrm{2}\right)\boldsymbol{{j}}+\left(\mathrm{2}\boldsymbol{{t}}^{\mathrm{3}} −\mathrm{4}\boldsymbol{{t}}^{\mathrm{2}} \right)\boldsymbol{{k}}. \\ $$$$\boldsymbol{{find}} \\ $$$$\left(\boldsymbol{{a}}\right)\boldsymbol{{velocity}} \\ $$$$\left(\boldsymbol{{b}}\right)\boldsymbol{{speed}}\:\boldsymbol{{or}}\:\boldsymbol{{magnitude}}\:\boldsymbol{{of}}\:\boldsymbol{{velocity}} \\ $$$$\left(\boldsymbol{{c}}\right)\boldsymbol{{acceleration}} \\…

Solve-The-position-vector-of-a-particle-at-any-time-t-is-given-by-r-acoswt-i-asinwt-j-bt-2-k-a-show-that-although-the-speed-of-the-particle-increases-with-time-the-magnitude-of-the-accelerati

Question Number 200737 by Calculusboy last updated on 22/Nov/23 $$\boldsymbol{{Solve}}:\:\boldsymbol{{The}}\:\boldsymbol{{position}}\:\boldsymbol{{vector}}\:\boldsymbol{{of}}\:\boldsymbol{{a}}\:\boldsymbol{{particle}}\:\boldsymbol{{at}}\:\boldsymbol{{any}}\:\boldsymbol{{time}}\:\boldsymbol{{t}} \\ $$$$\boldsymbol{{is}}\:\boldsymbol{{given}}\:\boldsymbol{{by}}\:\:\underset{−} {\boldsymbol{{r}}}=\left(\boldsymbol{{acoswt}}\right)\boldsymbol{{i}}+\left(\boldsymbol{{asinwt}}\right)\boldsymbol{{j}}+\boldsymbol{{bt}}^{\mathrm{2}} \boldsymbol{{k}} \\ $$$$\left(\boldsymbol{{a}}\right)\:\boldsymbol{{show}}\:\boldsymbol{{that}},\boldsymbol{{although}}\:\boldsymbol{{the}}\:\boldsymbol{{speed}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{particle}} \\ $$$$\boldsymbol{{increases}}\:\boldsymbol{{with}}\:\boldsymbol{{time}},\boldsymbol{{the}}\:\boldsymbol{{magnitude}} \\ $$$$\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{acceleration}}\:\boldsymbol{{is}}\:\boldsymbol{{always}}\:\boldsymbol{{constant}} \\ $$$$\left(\boldsymbol{{b}}\right)\:\boldsymbol{{describe}}\:\boldsymbol{{the}}\:\boldsymbol{{motion}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{particle}}\:\boldsymbol{{geometrically}} \\ $$ Terms…

Solve-A-smooth-sphere-A-of-mass-2kg-and-moving-with-speed-6ms-1-collides-obliquely-with-a-smooth-sphere-B-of-mass-4kg-just-before-the-impact-B-is-stationary-and-the-velocity-of-A-makes-an-angle-

Question Number 200738 by Calculusboy last updated on 22/Nov/23 $$\boldsymbol{{Solve}}:\:\boldsymbol{{A}}\:\boldsymbol{{smooth}}\:\boldsymbol{{sphere}}\:\boldsymbol{{A}},\boldsymbol{{of}}\:\boldsymbol{{mass}}\:\mathrm{2}\boldsymbol{{kg}}\:\boldsymbol{{and}} \\ $$$$\boldsymbol{{moving}}\:\boldsymbol{{with}}\:\boldsymbol{{speed}}\:\mathrm{6}\boldsymbol{{ms}}^{−\mathrm{1}} \boldsymbol{{collides}}\:\boldsymbol{{obliquely}}\: \\ $$$$\boldsymbol{{with}}\:\boldsymbol{{a}}\:\boldsymbol{{smooth}}\:\boldsymbol{{sphere}}\:\boldsymbol{{B}}\:\boldsymbol{{of}}\:\boldsymbol{{mass}}\:\mathrm{4}\boldsymbol{{kg}}.\:\boldsymbol{{just}}\:\boldsymbol{{before}}\:\boldsymbol{{the}}\:\boldsymbol{{impact}}\:\boldsymbol{{B}}\:\boldsymbol{{is}} \\ $$$$\boldsymbol{{stationary}}\:\boldsymbol{{and}}\:\boldsymbol{{the}}\:\boldsymbol{{velocity}}\:\boldsymbol{{of}}\:\boldsymbol{{A}}\:\boldsymbol{{makes}} \\ $$$$\boldsymbol{{an}}\:\boldsymbol{{angle}}\:\boldsymbol{{of}}\:\mathrm{10}°\:\boldsymbol{{with}}\:\boldsymbol{{the}}\:\boldsymbol{{lines}}\:\boldsymbol{{of}}\:\boldsymbol{{centers}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{two}}\:\boldsymbol{{sphere}}. \\ $$$$\boldsymbol{{The}}\:\boldsymbol{{coefficient}}\:\boldsymbol{{of}}\:\boldsymbol{{restitution}}\:\boldsymbol{{between}}\:\boldsymbol{{the}} \\ $$$$\boldsymbol{{spheres}}\:\boldsymbol{{is}}\:\frac{\mathrm{1}}{\mathrm{2}}.\:\boldsymbol{{Find}}\:\boldsymbol{{the}}\:\boldsymbol{{magnitude}}\:\boldsymbol{{and}}\: \\ $$$$\boldsymbol{{directions}}\:\boldsymbol{{of}}\:\boldsymbol{{the}}\:\boldsymbol{{velovities}}\:\boldsymbol{{of}}\:\boldsymbol{{A}}\:\boldsymbol{{and}}\:\boldsymbol{{B}}…

Find-all-polynomials-P-x-with-real-coefficients-such-that-for-all-nonzero-real-numbers-x-P-x-P-1-x-P-x-1-x-P-x-1-x-2-

Question Number 200722 by cortano12 last updated on 22/Nov/23 $$\:\:\mathrm{Find}\:\mathrm{all}\:\mathrm{polynomials}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{with} \\ $$$$\:\mathrm{real}\:\mathrm{coefficients}\:\mathrm{such}\:\mathrm{that}\:\mathrm{for} \\ $$$$\:\mathrm{all}\:\mathrm{nonzero}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x},\: \\ $$$$\:\:\:\:\:\mathrm{P}\left(\mathrm{x}\right)+\mathrm{P}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=\frac{\mathrm{P}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)+\mathrm{P}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{2}}\:\:\: \\ $$ Commented by Frix last updated on 22/Nov/23…

1-a-8-b-5-6-a-b-max-

Question Number 200716 by hardmath last updated on 22/Nov/23 $$\mathrm{1}.\: \\ $$$$\frac{{a}}{\mathrm{8}}\:+\:\frac{{b}}{\mathrm{5}}\:=\:\mathrm{6}\:\:\:\Rightarrow\:\:\:\left({a}+{b}\right)_{\boldsymbol{{max}}} \:=\:? \\ $$ Answered by AST last updated on 22/Nov/23 $$\mathrm{5}{a}+\mathrm{8}{b}=\mathrm{240}\Rightarrow{a}=\frac{\mathrm{240}−\mathrm{8}{b}}{\mathrm{5}} \\ $$$${remains}\:{to}\:{find}\:\left({a}+{b}\right)_{{max}}…