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SECTION A : The example in Feynman's Lectures Let a body P (Planet or Particle or whatever) moving in orbit around a center of attraction called $\:\rm{SUN}$, as in above Figure. Suppose that the attractive force $\:\mathbf{f}\left(r\right)\:$ depends continuously only on the distance $\:r\:$ of the body P from the center $\:\rm{SUN}$. Here it's not ...


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Energy kinematics I have this question, because typically problems that can be solve using conservation of energy or just energy-related principles, can usually be solved sing kinematic equations. Yep. In fact, there are two profound pieces of math, Hamiltonian and Lagrangian dynamics, which say that you can use energies to derive the actual kinematic ...


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The easiest way to calculate escape velocity, is neglicting Earths rotation and assuming the object takes of in a radial direction. Then, indeed, you start from $$E = K_1 + U_1 = K_2 + U_2$$ where $K_1=\frac{mv^2}{2}$ and $U_1=- \frac{GMm}{r}$. Since the range of gravitional forces is infinity, you say (theoretically, not practically) that an object has ...


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The point is that if $\frac{1}{2} mv^2 - GMm/r$ is constant, then $v$ only depends on $r$! This is surprising and very useful; it means that $v$ will be the same no matter what path a planet takes from some $r_1$ to $r_2$. In this case, the two paths he's using are the planet's usual elliptical orbit, and a path that goes straight toward the sun. You don't ...


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The experiment you link is Joule's classic paddle-wheel experiment. Specifically, Joule determined that applying 772.24 foot pound force via the weight produced a rise of 1 degree F in one pound of water, although later, more precise experiments gave slightly higher numbers. The experiment is described in exhaustive detail in Joule's paper, a copy of ...


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Energy and momentum are conserved at every vertex of a Feynman diagram in quantum field theory. No internal lines in a Feynman diagram associated with a virtual particles violate energy-momentum conservation. It is true, however, that virtual particles are off-shell, that is, they do not satisfy the ordinary equations of motion, such as $$E^2=p^2 + m^2.$$ ...


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SECTION A : Non-relativistic conservation of energy The work done by the non-relativistic force $\:\mathbf{f}\:$ per time unit, that is the power produced or consumed, on a particle moving with velocity $\:\mathbf{v}=d\mathbf{r}/dt\:$ is \begin{align} \dfrac{dW}{dt}=\mathbf{f}\circ \mathbf{v}=&\dfrac{d\mathbf{p}}{dt}\circ \mathbf{v}=\\ ...


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The energy gets converted into the form of heat and sound. In this way the energy is conserved.


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We may construct a system with Hamiltonian not $T+V$ but energy still conserved from any system where energy is conserved by making the phase space description generally covaraint: Starting from an unconstrained Hamiltonian $H_0(p,q)$ with Hamiltonian action $$ S_0 = \int \left(p_i \frac{q^i}{\mathrm{d}t} - H_0(p,q)\right)\mathrm{d}t$$ we may turn it into a ...


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It is not that easy. Now one engine is delivering about twice as much torque and was probably not designed for that. An engine specifically designed to deliver more torque would like be more efficient than the two combined. Lets look torque alone. Ignore the tow rope and assume the second is far enough back to not get any draft. In this case wind ...


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As noted in the comments, weight must be evenly distributed or the washing machine will spin off center and shut down. Clothes are a lot of small pieces. When spinning starts, they fly to the outside. Usually they are uniformly distributed. A duvet is a single large piece. It is easy for it to be off center. For example, if you wrap it around the ...


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Typically, the Friction force will be proportional to the velocity of the object it affects (or at least it is usually assumed to be proportional). The Force you describe is constant and pointing in negative x-direction. The situation you are describing resembles an object on a Hookean spring in a gravity field. At first, F1 pulls it upward (positive x) far ...


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Just before the ball reaches the ground, all of its molecules are coming down with almost an equal speed that is the speed of the ball.(Although, due to the non-zero temperature of the ball, the molecules are also vibrating about their mean position wrt COM frame of the ball).And thus the ball possesses a systematic macroscopic kinetic energy. Now when ...


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Your $E$ is potential energy in the rubber, which transforms to kinetic $K$. So your starting velocity $v$ will be: $E=1/2 mv^2$ From conservation of energy: $0-1/2 mv^2=0-mgh$ $h=v^2/2g$ and $v^2=2E/m$ Confirmed Interestingly enough the rubber does not obey Hook's law, and you need a lot more work if you want to find out what really ...


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A simple solution can be like this- At first you find the rate of change of Kinetic Energy. The process is as follows. $$d/dt(1/2 mv^2)$$ =$$F.v$$ Now imagine this case of a constant force acts on the body which equals to -mg(minus indicates downward direction). So, the equation is -mgv.Now the velocity is the rate of change of vertical position of the ...


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Assuming that the angle theta is measured relative to the vertical (e.g., the position of the string when the pendulum is at rest), a careful free body analysis indicates that the acceleration of the pendulum is g * sin(theta). This means that the acceleration of the pendulum continuously varies as it swings. This is relevant because the kinematic ...


1

You mean quantum tunneling? The particle doesn't really "borrow" energy, actually a particle will have a higher probability of tunneling through a barrier if it has a high kinetic energy. A naive analogy would be that the more energetic the bullet is, the higher the probability it has of piercing through a wall, that is, tunneling. What makes quantum ...


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As mentioned in the link you provided, it is due to Heisenberg's uncertainty relation $-$ during the short-lasting tunneling, the particle may temporarily borrow some energy from the potential of the barrier, so sometimes it can jump over it. Well, the energy and time may be depicted as a sort of Fourier transform pair (see Fourier transform) because the ...


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One should always remember that quantum mechanics predicts probabilities and not energy distributions . The energy a particle will have is an eigenvalue of the energy operator operating on the wavefunction, but the probability of finding a particle at (x,y,z) at time t is given by the complex conjugate square of the wavefunction which is the solution of the ...


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Feynman's trajectory The trajectory discussed by Feynman is shown below in red for the blue path, which is a hyperbolic deflection of a small particle around a large star centered at $(0, 0)$. Discussion Feynman's trajectory here trying to answer the question: how much has the speed increased between A and B. He is answering that by saying that there is ...


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You're right that the kinetic energy of the spacecraft is the same both before and after the planetary encounter—in the reference frame of the planet (or, technically, the frame of the planet-spacecraft CM.) But the fact that the kinetic energy is the same before & after in one frame does not mean that the kinetic energy will be the same before & ...


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It's just a way of saying that the S-matrix only connects initial states to final states that have the same energy and momentum. With finitely many states the S-matrix is a finite matrix, and the $m$:th element in the $n$:th row is non-zero if the time evolution of the $n$:th initial state has an $m$:th state component. In an energy conserving theory, it is ...


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Conservation of energy is exactly how the venturi effect arises. In an incompressible fluid, conservation of energy states that $E = E_k + E_{p,pressure} + E_{p,gravity} = \frac{1}{2} mv^2 + PV + mgz = constant$ by dividing each term with volume, it becomes the Bernoulli Principle: $\frac{1}{2} \rho v^2 + P_{static} + \rho gz = anotherConstant$ So, how ...


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I think one has to be very careful when talking about "particles popping in and out of existence". This interpretation is only sort of fine in flat-spacetime QFT, where the Minkowski metric is time-invariant, so has a global timeline Killing vector. The definition of a particle depends on the notion of there existing time invariance! Since black hole ...


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When a rocket is fired from Earth with a sudden impulse, its total energy is given by: $$E_k \text{ (kinetic energy)} + E_p \text{ (potential energy)}= \frac{1}{2}mv^2 - \frac{GMm}{r} = constant$$ The potential energy here is taken to be negative because the reference point chosen for potential energy to be zero is when the rocket is unbound in Earth's ...


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The experiment was designed to show that a mechanical process (a paddle wheel stirring water) could cause the water temperature to rise by a predictable amount. The equivalence was tested by using a system in which the mechanical work could be easily measured, a mass falling in a gravitational field. It turns out that there are several linear relationships ...



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