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$\bullet$ In an experiment, two protons are shot directly towardeach other, each moving at half the speed of light relative tothe laboratory. (a) What speed does one proton measure forthe other proton? (b) What would be the answer to part (a) ifwe used only nonrelativistic Newtonian mechanics? (c) Whatis the kinetic energy of each proton as measured by (i) anobserver at rest in the laboratory and (ii) an observer ridingalong with one of the protons? (d) What would be theanswers to part (c) if we used only nonrelativistic Newtonianmechanics?

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$$0 \mathrm{MeV}, 469.1 \mathrm{MeV}$$

Physics 101 Mechanics

Chapter 27

Relativity

Gravitation

Simon Fraser University

University of Sheffield

McMaster University

Lectures

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Okay, So in this problem we have on outsider observer in an observer in the laboratory, that's go reference frame s this absurd Watches to Broughton's collides with equal speed off. Let's call. This is Speed and this is Pete, you. So he is half of this beautiful light. Since this is since the speeds are factors, we need to consider the minor sign in front of you. And now we have the relation between into it speeds. Okay, So the first thing that problem wants to know is what is this, Pete, the relative speed between the two products. So always remember that he and you, our disputes relative to the inertial reference frame and re prime is the relative speed between the two more moving objects. So first of all, let's let's just remember to simplify the equation. The calculation that the rest mess off the is just let's see, MP there s too massive to broughtem is just one point 67 times 10 to the power of minus 27 kilograms and the rest energy off the problem, which is empty. C square is just 938 may have, so Okay, this is a nine. So let's calculate first the relative speed between you too Rot owns, and we know that to calculate the relative speed in knowing that the problems are under relativistic effects, we use the Lawrence Reformation for this speed. Ches v Minors You divided by one miner's UV divided by C Square, So this is equals two. See divided by two Pull us See divided by two divided by one well, us see spurned, divided by four c square. You can cut this season here so we have finally or see you've I did by five. And that's the relative speed between the two problems. If we calculate the relatives not using any relativistic effects using only did Newtonian mechanics, we know that the first problem as a Speed E and the second brother has a speed you. So without using the Lawrence Reformation, the relative Speed V Prime is just fi minors you. And since the they have opposite directions, this is just see divided by two plus C, divided by two equals C. That's the relative speed between the two problems. No, the third item. We have to calculate the relativistic connecting energy off the bra tone in two reference frame in the reference frame off the laboratory observer in in the reference frame off the proto itself. So let's calculate first to the laboratory frame, so we have that definition that you connect. The energy is just gamma M C square minus M c square. This is the definition of connecting energy and gamma, for this first reference frame is going to be Let's see the energy we have, which is my 138 divided by one minus e square. What is the vino is just see divided by two divided by sea this square Because this is the speech off the pro TEM measure by observer in the reference frame of the laboratory, we can cut this season here minus 938. So this is just 145 MEV. This is the connecting energy measure in the reference frame of the laboratory. Now let's do the same thing. But using this speech, the relative speed between the British in between the two products, this is just Keiko's gamma N C Square minus M C square. So this is going to be let's see K e Croce and 238. If I did buy one minus or divided by five square in this crab route, I forgot it's crab it here after Gamma minus 938. So after calculating this, we have that connecting energy is just 600 in 25. Matt. Now, finally, let's calculate your energy, but using the relativistic using the new 20 mechanics. So we have to calculate a connecting energy? No, using only the Newtonian mechanics using the Newtonian mechanics, we know that the connecting energy define is half M V square. So India old server in the laboratory, we see the speed as half of the speed of light, which means we have half mm C square divided by four. So calculating this this is just 938 divided by eight, which gives us 117 meth. This is the energy debt laboratory would measure. And, you know, Newtonian mechanics? No, the energy in the reference frame off the product. Um, again half of M V square. But this time the speed is if your prime we calculated before, which is just this beautiful light. So this is half of M V. C Square. So this is gives us 938 divided by two, which is 400 and 69 Meth. That's the final answer. Thanks for watching.

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