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free/or 0.5$ |
381 | A relativistic particle with momentum p and total energy E moves along the x axis of the frame K. Demonstrate that in the frame K' moving with a constant velocity V relative to the frame K in the positive direction of its axis x the momentum and the total energy of the given particle are defined by the formulas: px=.., E=.. where b = V/c. |
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382 | The photon energy in the frame K is equal to a. Making use of the transformation formulas cited in the foregoing problem, find the energy a' of this photon in the frame K' moving with a velocity V relative to the frame K in the photon's motion direction. At what value of V is the energy of the photon equal to e' = e/2? |
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383 | Demonstrate that the quantity E^2 — p^2c^2 for a particle is an invariant, i.e. it has the same magnitude in all inertial reference frames. What is the magnitude of this invariant? |
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384 | A neutron with kinetic energy T = 2m0c^2, where (a) the combined kinetic energy T of both neutrons in the frame of their centre of inertia and the momentum of each neutron in that frame; (b) the velocity of the centre of inertia of this system of particles. Instruction. Make use of the invariant E^2 — p^2c^2 remaining constant on transition from one inertial reference frame to another (E is the total energy of the system, p is its composite momentum). |
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385 | A particle of rest mass m0 with kinetic energy T strikes a stationary particle of the same rest mass. Find the rest mass and the velocity of the compound particle formed as a result of the collision. |
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386 | How high must be the kinetic energy of a proton striking another, stationary, proton for their combined kinetic energy in the frame of the centre of inertia to be equal to the total kinetic energy of two protons moving toward each other with individual kinetic energies T = 25.0 GeV? |
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387 | A stationary particle of rest mass m0 disintegrates into three particles with rest masses m1, m2, and m3. Find the maximum total energy that, for example, the particle m1 m ay possess. |
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388 | A relativistic rocket emits a gas jet with non-relativistic velocity u constant relative to the rocket. Find how the velocity v of the rocket depends on its rest mass m if the initial rest mass of the rocket equals m0. |
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389 | A vessel of volume V = 30 l contains ideal gas at the temperature 0 °C. After a portion of the gas has been let out, the pressure in the vessel decreased by Δp = 0.78 atm (the temperature remaining constant). Find the mass of the released gas. The gas density under the normal conditions ρ = 1.3 g/l. |
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390 | Two identical vessels are connected by a tube with a valve letting the gas pass from one vessel into the other if the pressure difference Δp ≥ 1.10 atm. Initially there was a vacuum in one vessel while the other contained ideal gas at a temperature t1 = 27 °C and pressure p1 = 1.00 atm. Then both vessels were heated to a temperature t2 = 107 °C. Up to what value will the pressure in the first vessel (which had vacuum initially) increase? |
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391 | A vessel of volume V = 20 l contains a mixture of hydrogen and helium at a temperature t = 20 °C and pressure p = 2.0 atm. The mass of the mixture is equal to m = 5.0 g. Find the ratio of the mass of hydrogen to that of helium in the given mixture. |
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392 | A vessel contains a mixture of nitrogen (m1 = 7.0 g) and carbon dioxide (m2 = 11 g) at a temperature T = 290 K and pressure p0 = 1.0 atm. Find the density of this mixture, assuming the gases to be ideal. |
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393 | A vessel of volume V = 7.5 1 contains a mixture of ideal gases at a temperature T = 300 K: n1 = 0.10 mole of oxygen, n2 = 0.20 mole of nitrogen, and n3 = 0.30 mole of carbon dioxide. Assuming the gases to be ideal, find: (a) the pressure of the mixture; (b) the mean molar mass M of the given mixture which enters its equation of state pV = (mIM) RT, where m is the mass of the mixture. |
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394 | A vertical cylinder closed from both ends is equipped with an easily moving piston dividing the volume into two parts, each containing one mole of air. In equilibrium at T0 = 300 K the volume of the upper part is η = 4.0 times greater than that of the lower part. At what temperature will the ratio of these volumes be equal to η' = 3.0? |
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395 | A vessel of volume V is evacuated by means of a piston air pump. One piston stroke captures the volume ΔV. How many strokes are needed to reduce the pressure in the vessel η times? The process is assumed to be isothermal, and the gas ideal. |
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396 | Find the pressure of air in a vessel being evacuated as a function of evacuation time t. The vessel volume is V, the initial pressure is p0. The process is assumed to be isothermal, and the evacuation rate equal to C and independent of pressure. Note. The evacuation rate is the gas volume being evacuated per unit time, with that volume being measured under the gas pressure attained by that moment. |
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397 | A chamber of volume V = 87 l is evacuated by a pump whose evacuation rate (see Note to the foregoing problem) equals C = 10 l/s. How soon will the pressure in the chamber decrease by η = 1000 times? |
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398 | A smooth vertical tube having two different sections is open from both ends and equipped with two pistons of different areas (Fig. 2.1). Each piston slides within a respective tube section. One mole of ideal gas is enclosed between the pistons tied with a non-stretchable thread. The cross-sectional area of the upper piston is ΔS = 10 cm2 greater than that of the lower one. The combined mass of the two pistons is equal to m = 5.0 kg. The outside air pressure is p0 = 1.0 atm. By how many kelvins must the gas between the pistons be heated to shift the pistons through l = 5.0 cm? |
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399 | Find the maximum attainable temperature of ideal gas in each of the following processes: (a) p = p0 - αV2; (b) p = p0e-βV, where p0, α and β are positive constants, and V is the volume of one mole of gas. |
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400 | Find the minimum attainable pressure of ideal gas in the process T = T0 + αV2, where T0 and α are positive constants, and V is the volume of one mole of gas. Draw the approximate p vs V plot of this process. |
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