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This is chapter 37 problem number 34.
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We're asked to calculate how much work needs to be done to accelerate a particle mass.
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From rest in part a.
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So we're calculating the work done, right? but the initial velocity is going to be zero in part a because it says from rest to a final velocity of 0 .09 % of the speed of light.
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Right so in order to calculate the work we know that work equals to the change in kinetic energy so in this case we need to calculate the kinetic energy that corresponds to initial speed of zero and the final speed of 0 .09c so then okay final final kinetic energy minus initial and kinetic energy is going to give us a delta k so let's write what initial kinetic energy would be of course relativistic right um so initial gamma minus 1 mc squared is going to be our initial kinetic energy.
00:59
Our final kinetic energy is going to be gamma final minus 1 mc square, right? so if we plug everything, these two in the work equation, we're going to have gamma final minus 1 mc squared minus, gamma initial minus 1 mc squared.
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So we have mc squared in both terms.
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So let's group the coefficients of these guys.
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So final minus 1 minus, minus.
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Initial minus 1, right? so if you do this, then we're going to get final minus gamma initial minus 1 plus 1, right? these are going to cancel.
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So as a final expression, we have mc squared final gamma minus initial gamma for the change in kinetic and that corresponds to work.
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Now, in order to calculate the final and initial gamma factors, we're going to have to plug in for them, right? which is the gamma factor is given as, this is the general formula, 1 minus the, well, whatever the corresponding velocity of that particle is over square of the speed of life, right? so we're going to write this for final gamma factor, and we're going to write it for initial gamma factor that corresponds to these two velocities.
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Now, let's go on a new page, then work done equals delta k.
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We said that equals final.
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Minus initial times mc squared.
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Final is going to be 1 minus v final squared over c squared minus 1 over square of 1 minus v initial squared over c squared.
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So times mc squared.
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Now, as you can see here, since the initial velocity is zero, this term is zero.
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So you're left with square of 1 here.
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Squared of 1 gives you 1.
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1 over 1 gives you 1.
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So this entire term actually equals to 1, right? so that's when that's the case, let's plug in the v final for here in the first term...