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So the problem is to find the trajectory of a particle under influence of gravity and air resistance quadratic air resistance. The governing equations, as they appear on the poster:

• He uses, right in the beginning, the substitution $\psi(t) = u(t)/v(t)$.

• There is a section called "...öße der Bewegung". The first word is obscured, but a qualified guess would be "Erhaltungsgröße der Bewegung", which would translate as "conserved quantity of the motion". Here, the conserved quantity described by David Zaslavsky appears, modulo some sign issues.

• However, this section seems to be a subsection to the bigger section "Aus der Lösung ablesbare Eigenschaften", or "Properties that can seen from the solution". That seems to imply that the solution implies the conservation law, rather than the solution being derived from the conservation law. The text in that section probably provides some clue, but it's only partly visible, and, well, my German is rusty. I welcome someone else to try to make sense of it.

• Also part of the bigger sections are subsections where he derives from his solution (a) the trajectory for classical, drag-free projectiles, (b) some "Lamb-Näherung", or "Lamb approximation".

• The next section is called "Verallgemeneirungen", or "Generalizations". Here, he seems to consider two other problems drag of the form $\alpha V^2 + \beta$, in the presence of altitude-dependent horizontal wind. I'm not sure what the results here are.

• The diagrams to the left seem to demonstrate the accuracy and convergence of his series solution by comparing them to Runge-Kutta. Though the text is kind of blurry, and, again, my German is rusty, so I'm not too sure.

• Here's a rough translation of the first part of the "Zusammanfassung und Ausblick" (Summary and outlook), with suitable disclaimers as to the accuracy:

There, the questions he solved are unambiguously stated, so that should answer any outstanding questions.

So the problem is to find the trajectory of a particle under influence of gravity and quadratic air resistance. The governing equations, as they appear on the poster:

• He uses, right in the beginning, the substitution $\psi(t) = u(t)/v(t)$.

• There is a section called "...öße der Bewegung". The first word is obscured, but a qualified guess would be "Erhaltungsgröße der Bewegung", which would translate as "conserved quantity of the motion". Here, the conserved quantity described by David Zaslavsky appears, modulo some sign issues.

• However, this section seems to be a subsection to the bigger section "Aus der Lösung ablesbare Eigenschaften", or "Properties that can seen from the solution". That seems to imply that the solution implies the conservation law, rather than the solution being derived from the conservation law. The text in that section probably provides some clue, but it's only partly visible, and, well, my German is rusty. I welcome someone else to try to make sense of it.

• Also part of the bigger section are subsections where he derives from his solution (a) the trajectory for classical, drag-free projectiles, (b) some "Lamb-Näherung", or "Lamb approximation".

• The next section is called "Verallgemeneirungen", or "Generalizations". Here, he seems to consider two other problems, with drag of the form $\alpha V^2 + \beta$, in the presence of altitude-dependent horizontal wind. I'm not sure what the results here are.

• The diagrams to the left seem to demonstrate the accuracy and convergence of his series solution by comparing them to Runge-Kutta. Though the text is kind of blurry, and, again, my German is rusty, so I'm not too sure.

• Here's a rough translation of the first part of the "Zusammanfassung und Ausblick" (Summary and outlook), with suitable disclaimers as to the accuracy:

There, the questions he solved are unambiguously stated, so that should answer any outstanding questions.

EDIT2: I should add: I do not doubt that Shouryya Ray is a very intelligent young man. The solution he gave can, perhaps, be obtained using standard methods. I believe, however, that he discovered the solution without being aware that the methods were standard, a very remarkable achievement indeed. I hope that this event has not discouraged him; no doubt, he'll be a successful physicist or mathematician one day, should he choose that path.

This thread(physicsforums.com) contains a link to Shouryya Ray's poster, in which he presents his results.

So the problem is to find the trajectory of a particle under influence of gravity and air resistance quadratic air resistance. The governing equations, as they appear on the poster:

$$ \dot u(t) + \alpha u(t) \sqrt{u(t)^2+v(t)^2} = 0 \\ \dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g\text, $$

subject to initial conditions $v(0) = v_0 > 0$ and $u(0) = u_0 \neq 0$.

Thus (it is easily inferred), in his notation, $u(t)$ is the horizontal velocity, $v(t)$ is the vertical velocity, $g$ is the gravitational acceleration, and $\alpha$ is a drag coefficient.

He then writes down the solutions

$$ u(t) = \frac{u_0}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots} \\ v(t) = \frac{v_0 - g\left[t + \tfrac{1}{2!} \alpha V_0 t^2 - \tfrac{1}{3!} \alpha gt^3 \sin \theta + \tfrac{1}{4!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right)t^4 + \cdots\right]}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots}\text.$$

From the diagram below the photo of Newton, one sees that $V_0$ is the inital speed, and $\theta$ is the initial elevation angle.

The poster (or at least the part that is visible) does not give details on the derivation of the solution. But some things can be seen:

• He uses, right in the beginning, the substitution $\psi(t) = u(t)/v(t)$.

• There is a section called "...öße der Bewegung". The first word is obscured, but a qualified guess would be "Erhaltungsgröße der Bewegung", which would translate as "conserved quantity of the motion". Here, the conserved quantity described by David Zaslavsky appears, modulo some sign issues.

• However, this section seems to be a subsection to the bigger section "Aus der Lösung ablesbare Eigenschaften", or "Properties that can seen from the solution". That seems to imply that the solution implies the conservation law, rather than the solution being derived from the conservation law. The text in that section probably provides some clue, but it's only partly visible, and, well, my German is rusty. I welcome someone else to try to make sense of it.

• Also part of the bigger sections are subsections where he derives from his solution (a) the trajectory for classical, drag-free projectiles, (b) some "Lamb-Näherung", or "Lamb approximation".

• The next section is called "Verallgemeneirungen", or "Generalizations". Here, he seems to consider two other problems drag of the form $\alpha V^2 + \beta$, in the presence of altitude-dependent horizontal wind. I'm not sure what the results here are.

• The diagrams to the left seem to demonstrate the accuracy and convergence of his series solution by comparing them to Runge-Kutta. Though the text is kind of blurry, and, again, my German is rusty, so I'm not too sure.

• Here's a rough translation of the first part of the "Zusammanfassung und Ausblick" (Summary and outlook), with suitable disclaimers as to the accuracy:

• For the first time, a fully analytical solution of a long unsolved problem
• Various excellent properties; in particular, conserved quantity $\Rightarrow$ fundamental [...] extraction of deep new insights using the complete analytical solutions (above all [...] perspectives and approximations are to be gained)
• Convergence of the solution numerically demonstrated
• Solution sketch for two generalizations

This thread(physicsforums.com) contains a link to Shouryya Ray's poster, in which he presents his results.

So the problem is to find the trajectory of a particle under influence of gravity and air resistance quadratic air resistance. The governing equations, as they appear on the poster:

$$ \dot u(t) + \alpha u(t) \sqrt{u(t)^2+v(t)^2} = 0 \\ \dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g\text, $$

subject to initial conditions $v(0) = v_0 > 0$ and $u(0) = u_0 \neq 0$.

Thus (it is easily inferred), in his notation, $u(t)$ is the horizontal velocity, $v(t)$ is the vertical velocity, $g$ is the gravitational acceleration, and $\alpha$ is a drag coefficient.

He then writes down the solutions

$$ u(t) = \frac{u_0}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots} \\ v(t) = \frac{v_0 - g\left[t + \tfrac{1}{2!} \alpha V_0 t^2 - \tfrac{1}{3!} \alpha gt^3 \sin \theta + \tfrac{1}{4!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right)t^4 + \cdots\right]}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots}\text.$$

From the diagram below the photo of Newton, one sees that $V_0$ is the inital speed, and $\theta$ is the initial elevation angle.

The poster (or at least the part that is visible) does not give details on the derivation of the solution. But some things can be seen:

• He uses, right in the beginning, the substitution $\psi(t) = u(t)/v(t)$.

• There is a section called "...öße der Bewegung". The first word is obscured, but a qualified guess would be "Erhaltungsgröße der Bewegung", which would translate as "conserved quantity of the motion". Here, the conserved quantity described by David Zaslavsky appears, modulo some sign issues.

• However, this section seems to be a subsection to the bigger section "Aus der Lösung ablesbare Eigenschaften", or "Properties that can seen from the solution". That seems to imply that the solution implies the conservation law, rather than the solution being derived from the conservation law. The text in that section probably provides some clue, but it's only partly visible, and, well, my German is rusty. I welcome someone else to try to make sense of it.

• Also part of the bigger sections are subsections where he derives from his solution (a) the trajectory for classical, drag-free projectiles, (b) some "Lamb-Näherung", or "Lamb approximation".

• The next section is called "Verallgemeneirungen", or "Generalizations". Here, he seems to consider two other problems drag of the form $\alpha V^2 + \beta$, in the presence of altitude-dependent horizontal wind. I'm not sure what the results here are.

• The diagrams to the left seem to demonstrate the accuracy and convergence of his series solution by comparing them to Runge-Kutta. Though the text is kind of blurry, and, again, my German is rusty, so I'm not too sure.

• Here's a rough translation of the first part of the "Zusammanfassung und Ausblick" (Summary and outlook), with suitable disclaimers as to the accuracy:

• For the first time, a fully analytical solution of a long unsolved problem
• Various excellent properties; in particular, conserved quantity $\Rightarrow$ fundamental [...] extraction of deep new insights using the complete analytical solutions (above all [...] perspectives and approximations are to be gained)
• Convergence of the solution numerically demonstrated
• Solution sketch for two generalizations

EDIT: Two professors at TU Dresden, who have seen Mr Ray's work, have written some comments:

Comments on some recent work by Shouryya Ray

There, the questions he solved are unambiguously stated, so that should answer any outstanding questions.

This thread(physicsforums.com) contains a link to Shouryya Ray's poster, in which he presents his results.

So the problem is to find the trajectory of a particle under influence of gravity and air resistance quadratic air resistance. The governing equations, as they appear on the poster:

$$ \dot u(t) + \alpha u(t) \sqrt{u(t)^2+v(t)^2} = 0 \\ \dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g\text, $$

subject to initial conditions $v(0) = v_0 > 0$ and $u(0) = u_0 \neq 0$.

Thus (it is easily inferred), in his notation, $u(t)$ is the horizontal velocity, $v(t)$ is the vertical velocity, $g$ is the gravitational acceleration, and $\alpha$ is a drag coefficient.

He then writes down the solutions

$$ u(t) = \frac{u_0}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots} \\ v(t) = \frac{v_0 - g\left[t + \tfrac{1}{2!} \alpha V_0 t^2 - \tfrac{1}{3!} \alpha gt^3 \sin \theta + \tfrac{1}{4!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right)t^4 + \cdots\right]}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots}\text.$$

From the diagram below the photo of Newton, one sees that $V_0$ is the inital speed, and $\theta$ is the initial elevation angle.

The poster (or at least the part that is visible) does not give details on the derivation of the solution. But some things can be seen:

• He uses, right in the beginning, the substitution $\psi(t) = u(t)/v(t)$.

• There is a section called "...öße der Bewegung". The first word is obscured, but a qualified guess would be "Erhaltungsgröße der Bewegung", which would translate as "conserved quantity of the motion". Here, the conserved quantity described by David Zaslavsky appears, modulo some sign issues.

• However, this section seems to be a subsection to the bigger section "Aus der Lösung ablesbare Eigenschaften", or "Properties that can seen from the solution". That seems to imply that the solution implies the conservation law, rather than the solution being derived from the conservation law. The text in that section probably provides some clue, but it's only partly visible, and, well, my German is rusty. I welcome someone else to try to make sense of it.

• The next section is called "Verallgemeinerungen", or "Generalizations". Here, he seems to consider two other problems drag of the form $\alpha V^2 + \beta$, in the presence of altitude-dependent horizontal wind. I'm not sure what the results here are.

• The diagrams to the left seem to demonstrate the accuracy and convergence of his series solution by comparing them to Runge-Kutta. Though the text is kind of blurry, and, again, my German is rusty, so I'm not too sure.

This thread(physicsforums.com) contains a link to Shouryya Ray's poster, in which he presents his results.

So the problem is to find the trajectory of a particle under influence of gravity and air resistance quadratic air resistance. The governing equations, as they appear on the poster:

$$ \dot u(t) + \alpha u(t) \sqrt{u(t)^2+v(t)^2} = 0 \\ \dot v(t) + \alpha v(t) \sqrt{u(t)^2 + v(t)^2} = -g\text, $$

subject to initial conditions $v(0) = v_0 > 0$ and $u(0) = u_0 \neq 0$.

Thus (it is easily inferred), in his notation, $u(t)$ is the horizontal velocity, $v(t)$ is the vertical velocity, $g$ is the gravitational acceleration, and $\alpha$ is a drag coefficient.

He then writes down the solutions

$$ u(t) = \frac{u_0}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots} \\ v(t) = \frac{v_0 - g\left[t + \tfrac{1}{2!} \alpha V_0 t^2 - \tfrac{1}{3!} \alpha gt^3 \sin \theta + \tfrac{1}{4!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right)t^4 + \cdots\right]}{1 + \alpha V_0 t - \tfrac{1}{2!}\alpha gt^2 \sin \theta + \tfrac{1}{3!}\left(\alpha g^2 V_0 \cos^2 \theta - \alpha^2 g V_0 \sin \theta\right) t^3 + \cdots}\text.$$

From the diagram below the photo of Newton, one sees that $V_0$ is the inital speed, and $\theta$ is the initial elevation angle.

The poster (or at least the part that is visible) does not give details on the derivation of the solution. But some things can be seen:

• He uses, right in the beginning, the substitution $\psi(t) = u(t)/v(t)$.

• There is a section called "...öße der Bewegung". The first word is obscured, but a qualified guess would be "Erhaltungsgröße der Bewegung", which would translate as "conserved quantity of the motion". Here, the conserved quantity described by David Zaslavsky appears, modulo some sign issues.

• However, this section seems to be a subsection to the bigger section "Aus der Lösung ablesbare Eigenschaften", or "Properties that can seen from the solution". That seems to imply that the solution implies the conservation law, rather than the solution being derived from the conservation law. The text in that section probably provides some clue, but it's only partly visible, and, well, my German is rusty. I welcome someone else to try to make sense of it.

• Also part of the bigger sections are subsections where he derives from his solution (a) the trajectory for classical, drag-free projectiles, (b) some "Lamb-Näherung", or "Lamb approximation".

• The next section is called "Verallgemeneirungen", or "Generalizations". Here, he seems to consider two other problems drag of the form $\alpha V^2 + \beta$, in the presence of altitude-dependent horizontal wind. I'm not sure what the results here are.

• The diagrams to the left seem to demonstrate the accuracy and convergence of his series solution by comparing them to Runge-Kutta. Though the text is kind of blurry, and, again, my German is rusty, so I'm not too sure.

• Here's a rough translation of the first part of the "Zusammanfassung und Ausblick" (Summary and outlook), with suitable disclaimers as to the accuracy:

• For the first time, a fully analytical solution of a long unsolved problem
• Various excellent properties; in particular, conserved quantity $\Rightarrow$ fundamental [...] extraction of deep new insights using the complete analytical solutions (above all [...] perspectives and approximations are to be gained)
• Convergence of the solution numerically demonstrated
• Solution sketch for two generalizations