How do you build personas based on real users. Georgia doing "hand recount" of 2020 Presidential Election Ballots. I came across it as a question in an older A level M2 textbook by a remarkably inventive author D. Quadling . To Show that two complementary angles of projections give the same range of the projectile. rev 2020.11.12.37996, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, $θ = \arctan \left(\frac u{\sqrt{u^2+2gH}} \right)$. $$
Does the law of reflection hold for balls bouncing inelastically on inclined planes? What are recommended ways to connect fridge ice maker? endstream Solving for $\frac{dx}{d\theta}$ yields Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. MathJax reference. Lighting network what is the difference between wallet balance, local balance and remote balance? 2 0 obj << Since we are looking for the maximum range we set $y=0$ (i.e. /Parent 12 0 R Making statements based on opinion; back them up with references or personal experience. �Y�I����t�Qd��꧟y��E(L:�=��{P�Y�jKigN�bs,D��f�J�w�O���r��2�����-,�?��Z8� H + x\tan(\theta) - \frac1{2L}x^2(1+\tan^2(\theta)) = 0
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Your answer does not explain why the angle is what it is. What is the reason for the date of the Georgia runoff elections for the US Senate? $$
If we let $L=u^2/g$, then Our projectile motion calculator is a tool that helps you analyze the parabolic projectile motion. 1 Range of Projectile Motion 1.1 Horizontal Range Most of the basic physics textbooks talk about the horizontal range of the projectile motion. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. x��ZK��6����mյc_�c��RImR�=$����C&�����-Dz����/@P�$���>N�H�@������?��3o����3���e�[&d��-�_�Uu;S���r���>�f}�Y�z[�*����n��� ��~ IG�mV ��+H�2@� 9�*�*~.,.�΄����̳d�J���*�`���Zt����\h2�8�Y?QC�3� In all these "extreme" conditions your equation seems to work; but it definitely doesn't mean that it has to be correct. What is the optimal voltage for storing a 3.7V LiPo battery? Thanks for contributing an answer to Physics Stack Exchange! $$ By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Also, I can't see why it is not an intuitive reason, as the result has to be properly demonstrated with the standard kinematics equations. What does it mean to treat space and time on equal footing? Here's a scan of where I got the problem from: We are given the trajectory of a projectile: H + L - \frac{1}{2L}x^2 - \frac{L}2 = 0. $$ Category theory and arithmetical identities. Ask Question Asked 2 years, 11 months ago. Content Times: 0:16 Defining Range To subscribe to this RSS feed, copy and paste this URL into your RSS reader. where $H$ is the initial height, $g$ is the (positive) gravitational constant and $u$ is the initial speed. Was AGP only ever used for graphics cards? >> ... What’s the range for your cannon if you aim it at 45 degrees, which gives you your maximum range? $\endgroup$ – Patrick Apr 3 … /ProcSet [ /PDF /Text ] /Resources 1 0 R Why does a flight from Melbourne to Cape Town need to go via Doha? 3 0 obj << I already derived the angle of $\theta$ which maximises the distance on the inclined plane which turns out to be $\theta = \frac{\pi}{4} - \frac{\alpha}{2}$. It pretty obvious in the setting of the problem. /Length 2630 Finally, we replace $\tan(\theta)$ with $\frac{L}{x}$ in the second equation from the top and solve for $x$. =\frac {\frac {u^2}g}{\frac ug \sqrt{u^2+2gH}}=\color{red}{\frac u{\sqrt{u^2+2gH}}}$$ This derivative is $0$ when $\tan(\theta) = \frac{L}{x}$ and hence this corresponds to a critical number $\theta$ for the range of the projectile. We should now show that the $x$ value it corresponds to is a maximum, but I'll just assume that's the case. If a projectile is launched at a speed $u$ from a height $H$ above How were the cities of Milan and Bruges spared by the Black Death? Hence range and maximum height are equal for all bodies that are thrown with the same velocity and direction. Differentiate both sides with respect to $\theta$. Making statements based on opinion; back them up with references or personal experience. rev 2020.11.12.37996, The best answers are voted up and rise to the top, Physics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, Some proofs which rely only on geometry in the velocity co-ordinate system (with axes parallel and perpendicular to the incline) are given in. /MediaBox [0 0 595.276 841.89] Use MathJax to format equations. where $\ell$ is the linear distance between the launch and end points of the projectile. I think this is what you wanted to see. then differentiating with respect to $\theta$ so that we can let $\frac {dR}{d\theta}=0$ (so that $R=R_{\max }$), but this would eliminate the $H$, so it won't lead to the expression for $R_{\max }$ that I want to derive. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Maybe you could show one ? 3 $\begingroup$ A Ball is thrown up an inclined plane (incline angle $\alpha$) with $\vec{v_0}$ being at an angle of $\theta$ with the inclined plane. I confess that I used the brute force differentiation method to get through it but realised that Quadling had laid out the problem with a strong hint that a rather lovely simple solution was possible without calculus. and What is the lowest level character that can unfailingly beat the Lost Mine of Phandelver starting encounter? This answer merely restates the result already given in the question. %���� Use MathJax to format equations. Different results for the Hamiltonian of a disc rolling on an inclined plane, Lagrangian method applied to mass in inclined plane. Active 2 years, 11 months ago. Adapting concepts from the question and solutions here, we have. Anyway, downvoting a correct answer that was accepted by the OP is quite useless. Are bleach solutions still routinely used in biochemistry laboratories to rid surfaces of bacteria, viruses, certain enzymes and nucleic acids? Asking for help, clarification, or responding to other answers. This also seems to be correct when I compare it to other results on the internet. Range of the Projectile, R: The range of the projectile is the displacement in the horizontal direction. >> $$
The angle $\theta$ can now be found easily. Step 1: Now we are given initial velocity with which projectile is launched.