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1

As Andres Salas said, your kinetic energy should be the negative of the potential energy at the surface. Also, you're using the wrong formula for the potential energy in the crater. $-GMm/r$ is correct only when $r\ge R$.


1

The answer is quite simple. You can't see it because you've forgotten that you make an unphysical simplification when considering "tied to an immovable wall" type situations. What's unphysical about the situation is quite simple : there's no such thing as an immovable wall. Let's say the rocket is tied to the earth. To say where the lost chemical potential ...


0

The rocket motor generates thrust by acelerating the exhaust gases that it emits. The force on the rocket is equal to the change in momentum of the exhaust gases per second, i.e. the exhaust velocity times the exhaust mass per second. For the rocket travelling in space the energy generated from burning the fuel goes partly into the kinetic energy of the ...


0

If you assume that the fluid is incompressible, I'm relatively sure that it could be shown that 100% efficiency is theoretically possible. You could use a variety of mental models to do this. I would prefer to think about conventional hydraulics lifting some weight. If you match the weight to the pressure, then you could raise an external object some (m g h) ...


0

Turbines (impellers) have fins that work more like aircraft wings and sails than the buckets of a old water wheel. They have have very high lift/drag ratios which translate the high efficiency. It is friction and heat loss that kill efficiency not the work of the turbine (how much it spins).


2

Newton's third law. For the force the magnet exerts on the metal, there must be an equal an opposite force on the magnet exerted by the metal. Since both form one system (metal + truck + magnet), the net force on the system is zero, and it won't move.


1

If your electron is in a pure state then it's an eigenfunction, $\psi_e$, of the Hamiltonian describing it, $H_e$. The measuring system will also, in principle at least, be described by some wavefunction, $\psi_m$. If the two don't interact then the total wavefunction will just be a product: $$ \Psi = \psi_e\psi_m $$ and the system won't change with time. ...


2

When you measure the position of an electron that is in a pure energy state, what happens the energy becomes non-deterministic. An electron in a pure energy state is in a bound state. To "measure it" you have to excite it or , if it is in an already excited state measure the photon of its deexcitation. You cannot measure its position, while bound, to ...


1

If you're looking for a general solution to the schrodinger equation then yes, it is possible for the atom to be in a superposition of energy states. This does not violate conservation of energy. Can you see why? It is a subtle point. To start you off -- how do you measure the position of the electron in the first place? You must hit it with something. This ...


0

It is incorrect to state that The magnet is only braked by the friction of the axis and the air. It is not clear to me exactly what you're proposing, but whenever you deliver electrical energy to some external system you will slow down the rotation of the coil. This is usually through an inductive torque caused on the rotating coil by the currents that ...


0

It seems to me the obvious answer is that it agrees with all the experiments. I am an experimentalist after all, (the neutrino is perhaps the best "missing energy" story.)


0

Chemical reactions satisfy mass conservation and energy conservation independently. Mass conservation implies the mass of the system remaining unchanged (zero change) as the reaction proceeds. Small changes - however small - are deemed to violate the law of conservation of mass. This has nothing to do with our ability to measure very small values of mass ...


2

When the universe expands, it is important to understand that how its energy content evolves depends on the form of energy involved. If all that energy is locked up in the form of mass energy, then the density of that matter will decrease proportionally to the relative increase of any arbitrary volume of the universe (i.e. if expansion doubles the size of ...


6

Temperature means energy. The heat energy is still here. It's just that the "object" (the Universe) grown bigger so this energy had to spread through it. The more energy in a single point, the hotter it is. That's why they say it got cooler. It's like the expanding gas from your spray deodorant is cold when it leaves the can, but it was at room temperature ...


0

(Not an answer as such, more an extended comment and suggestion). You wrote "On the other hand, the 1-D elastic collision problem can't be solved with Conservation of Momentum alone - Conservation of Energy is also required. " Actually the 1D elastic collision problem can be solved with (i) Conservation of Momentum (COM) (ii) application of symmetry (iii) ...


1

The efficiency of large hydroelectric generators can be very high - up to 95% in ideal cases - however the efficiency of small installations is a lot lower and in particular it's hard to get efficient electricity generation if the flow rate is low, which is likely to be the case for your friend. But lets see what the potential is. You don't say what the ...


1

You may doubt that energy is conserved, but it is a direct consequence of Noether's theorem together with the assumption of time translation invariance, and this latter assumption is perhaps a bit more palatable/fundamental. That is, it is mathematically true that if the outcome of an experiment doesn't depend on when we perform it, the quantity we call ...


3

Let's assume that an experiment is performed that seems to indicate a violation of the conservation of energy principle. Now, I suppose that it's logically possible that the experiment actually and unambiguously falsifies the principle in which case we must conclude that the principle is approximate and we must seek a deeper principle to guide our ...


4

I will try to give a short introduction into the ideas of scientific truth as I understand them. In mathematics, the world is beautifully simple. We have axioms that the set to be true, and from these we can deduce a plethora of statements to be undoubtedly true - given that the axioms are true. There may be undecidable statements about which we cannot say ...


0

It cannot be demostrated, but it can be checked. This year I've done an experiment which verifies (indirectly) this principle. It is really easy: I take an iron ball, and I shoot it to a pendulum. As a result, the pendulum goes up, and I can measure the deviation angle after the collision. With a sensor, I'm also able to measure the initial speed of the ...


3

I will make my comments into an answer: If we are talking of the classical domain, energy is conserved but can change forms. Inelastic scattering has a strict definition and is usually used describing two body scattering processes. In two body impact situations conservation of input energy in the output products would require that the target has some ...


0

If you are able to reduce friction to a minimum, but not to zero, may be able to run your machine for a long time, but not perpetually. Any energy loss must eventually be offset by the introduction of extra energy into the system. If you're really interested in running forever, there's no such thing as an insignificant energy loss.


9

Yes, when you fire a pistol the hammer hits the bullet with a relatively small initial kinetic energy but the kinetic energy of the hammer and bullet after the collision is considerably higher. This may seem a silly example, but I think it actually highlights the important principle involved. In general when two bodies undergo an inelastic collision part of ...


0

Dark energy due to a cosmological constant does not get diluted by metric expansion of space. However, this does not violate energy conservation as the increase in energy will be cancelled by gravitational potential energy. The problem with general relativity is (some would say arguably) not energy conservation, but energy localization: In a rotating frame ...


2

The cosmological constant is a constant energy density per unit volume of space, so as the universe expands this does indeed create energy as it creates new space. In this sense conservation of energy is violated. Actually this is less surprising than you might think. Conservation of energy is linked to a symmery called time shift symmetry by Noether's ...


1

Actually, energy is often not Conserved in general Relativity. For are more in deep explanation see: http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html But just notice that Dark Energy might not necessarily end up being the cosmological constant, but a new force field, so its behavior might differ from that of an actual cosmological ...


0

The answer to the second question is actually quite straightforward: by computing $\partial_t E$ and using what given in the Gross-Pitaevskii for $\dot{\psi}$ and $\dot{\psi^*}$ one can check that all the terms cancel out so that $\partial_tE=0$. To derive the expression for the energy one could also start considering a lagrangian giving the ...


1

In general, the elasticity of a collision is dependent on the properties of the colliding objects. In a perfectly elastic collision, no kinetic energy is dissipated, which means the collision creates no heat, no sound, etc. In a perfectly inelastic collision, the maximum possible amount of kinetic energy is dissipated as heat, sound, etc. This corresponds ...



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