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$\bullet$ Sketch a graph of (a) the nonrelativistic Newtonianmomentum as a function of speed $v$ and (b) the relativisticmomentum as a function of $v .$ In both cases, start from $v=0$and include the region where $v \rightarrow c .$ Does either of thesegraphs extend beyond $v=c ?$

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(b) For non-relativistic motion, momentum $p=\frac{p=\frac{m v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$When $v$ is small, then $p=m v$ so the graph is linear. But when $v$ is comparable to $c$ then $p$ is notlinear. So the graph is as shown in the figure below.

Physics 101 Mechanics

Chapter 27

Relativity

Gravitation

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University of Michigan - Ann Arbor

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Okay, So in this problem, we want to sketch a graph for the no relativistic mo mentum of a particle and for the relativistic mo mental of a particle as a function off this speech. So, first of all, we know that the known relativistic mo mental of a particle is just envy. So since this is a linear function, we can see that the graph off the mo mentum off this particle should just be a straight line. So that's putting here a straight line that begins in zero. Is Alito accord in here? But it should begin in the zero. Because when the equal zero, we know that the Momenta MME should also be zero. And here we have. This is a graph off his speed. Mo mentum versus the speed. Here we have M c. And here we have the speed of light that is sick. So this point and here it should be the limit off the mo mentum of article that can can is ext. But we can see that the Mo mentum goes beyond this limit and that's a problem. But let's not think about this right now. That's feet first, I think about the graph for and no relativistic mo mentum. We know that the real activist IQ mo mentum sorry should be gamma envy. So we know that the relativistic mo mentum is just and V divided by one minus be square c square in the square roots But this is a complicated graft to think about and do a sketch. But let's think about all the in the denominator. Gamma, We know that gamma is just function off the velocity square in the square root actually gamma minus one. So we know this kind of function of a scary square root should be plot like this. And if we're invert dysfunction Gamma being one divided by one minus V square, it should also be inverted. So now our function we'll begin in this zero and go way up. And here we have a limit. This limit cannot be cross. And if we come back here in the momento, the relativistic Momenta, this limit is just they speed off the light. It's a foot, hear and see. So if we make what we can see there when the velocity is zero, the relativistic mo mentum should also be zero. So he begins in here and go way up, never cross the limit. And that's the difference between the relativistic mo Mental and the No relativistic momento. But a mystic Momenta has no limits, but the relativistic has the limit. That is the speed of light. So that's the answer. Thanks for watching.

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