I can't understand where I am making mistake?

The qn is

A standing man, observes rain falling with velocity of 20 m/sat an angle of 30° with the vertical. Now if he further increases his speed, rain again appears to fall at 30° with the vertical. Find his new velocity.

I made a practice test for junior year physics and dynamic, so if anyone could use the paper before their test, do it for fun, or review the questions and the test structure that would be great. Feedback as well please.

In class, I learned the proof for the conservation of mechanical energy of a simple pendulum using scalar values, as shown in the picture I attached.

However, I was wondering if it's possible to prove this rigorously by calculating the work done using the net force vector and the instantaneous displacement vector.

Here is what's confusing me:

1. The pendulum moves along a curved path, so the direction of the displacement vector changes at every single instant.
2. The net force vector (which is the vector sum of gravity and tension) is almost never parallel to the instantaneous displacement vector.

Despite these complexities, is it mathematically possible to prove it this way? Does the dot product naturally take care of the curved path and the non-parallel forces (like tension doing zero work)?

I would really appreciate it if someone could explain how the math works out or show the derivation! Thanks!

Please can you explain why we take the gravitational potential energy as negative when finding the total energy(subtract it)? How come some questions we ignore the negative? What does the negative answer mean?
Thanks!

Hi. Pleasure to meet everyone, can I please get assist anywhere I tried ai and kept getting different answers I'm not sure whether I'm doing something wrong all I'm asking is for help for this question

There is like zero tolerance around it so prying is probably not an option.

Edit: I solved it, just pit it in the freezer for an hour with an ice cube inside and the marble shrunk and came out super easy, no hot water needed.

A visually stunning animation of Harmonograph patterns—geometric drawings created by combining perpendicular oscillations.

For more click www.instagram.com/craftsandengineering

Quantum Physics Series

Video 1 of 6: Quantum Electrodynamics visualization using Feynman Diagrams

Author: Mugambi Ndwiga
In: www.instagram.com/craftsandengineering

This animation visualizes the fundamental interactions of Quantum Electrodynamics (QED) using Feynman diagram conventions. QED is the relativistic quantum field theory of electrodynamics, describing how light and matter interact.

Visualized Phenomena

The animation cycles through six key physical processes:

1. Compton Scattering: A photon hits an electron, resulting in an energy shift and change in direction.
2. Electron-Positron Annihilation: An electron and its antiparticle (positron) collide to produce high-energy photons ().
3. Pair Production: A high-energy photon interacts with the electric field of an atomic nucleus to create an electron-positron pair.
4. Bremsstrahlung (Braking Radiation): A charged particle (electron) is deflected by a nucleus and radiates energy as a photon.
5. Møller Scattering: The interaction and repulsion between two electrons via the exchange of a virtual photon.
6. Vacuum Polarization: A process where a photon temporarily fluctuates into a virtual electron-positron pair, affecting the vacuum's permittivity.

For code and more click Mathematical-video-animations-and-visualization/QED_Feynman_Diagrams_Animations.ipynb at main · zombimann/Mathematical-video-animations-and-visualization