At what speed do the relativistic formulas for (a) length and ...

Question

At what speed do the relativistic formulas for (a) length and (b) time intervals differ from classical values by 1.00%? (This is a reasonable way to estimate when to use relativistic calculations rather than classical.)

Accepted Answer

Welcome back. Everyone in this problem. A spacecraft travels at a high velocity relative to earth. At what fraction of the speed of light does the spacecraft's time dilation cause a clock on the spacecraft to tick 2% slower than a clock on earth and for the length to contract by 2% compared to its rest length. A says that for a 2% slower tick, the speed is 0.12 C. While for a 2% shorter length, the tick is the speed. Sorry is 0.197 C B says they are 0.12 and 0.199 C respectively. C 0.197 C for both and D 0.197 and 0.199 C respectively. Now let's try to solve by first thinking about the speed for the time dilation that causes our clock on the spacecraft to tick 2% slower than a clock on earth. What do we know about the time dilation? Well, recall, OK, that by the time dilation formula, the, the time interval in earth's frame delta T equals the time interval in the spacecraft frame, delta T not divided by the square root of one minus V squared divided by C squared. And now from this problem, we are trying to solve for V. But what do we know about our times? Well, for a 2% slower tick, that means delta T is going to be equal to 1.02 multiplied by delta T knot. OK. So now if we substitute this into our formula, we should be able to solve for V because no, this means 1.02 delta T knot equals delta T knot divided by the square root of one minus V squared divided by C squared. And if we cancel delta T knot from both sides, then we get 1.02 to be equal to one divided by the square root of one minus V squared divided by C squared. Let's do some transposition to help us solve for V. Now, in this case, then that means that uh the square root of one minus V squared divided by C squared is going to be equal to one divided by 1.02. And know if we square both sides and make V squared divided by C squared subject, then V squared divided by C squared equals one minus one divided by 1.02 squared does V squared equals C squared multiplied by this expression. OK. And now to get V, we can square root both sides where V is going to be equal to C multiplied by the square root of one minus one divided by 1.02 squared. Now, when we go ahead and solve here to, to, to, to solve for V by putting this into our calculator, then we should get V to be equal to 0.1971 multiplied by C or to three significant figures. Approximately 0.197 C. Thus for a 2% slower tick, then our speed is going to be 0.197 C approximately. Now, let's see if we can go ahead and figure out the next part of the problem for a contraction of 2% in length. Let's call that part two. Let's put that in blue. OK? Now, what do we know about our length contraction here? Well, we also know you're in this problem where L is gonna be equal to 0.98 multiplied by L knot where L is the proper length. Sorry L knot is the proper length and L is the contracted length. We also know that our contracted length is equal to the proper length divided by the Lawrence factor gamma. OK. So since we have this expression for L 0.98 L knot, that means 0.98 L knot is equal to LK, not divided by gamma, which means that 0.98 equals one divided by gamma. But what do we know about the lines factor. Well, recall that the lines factor is equal to OK. The lines factor is equal to one divided by the square root of one minus V squared divided by C squared. OK. So now what this means then is that from our problem, we should find that one divided by or rather, you know, let me fix this here. I meant to. Well, no, you know what, let's keep it, my apologies, everyone. Let's continue. So if we compare both terms, then this tells us, OK, that 0.98 OK is going to be equal to the square root of one minus V squared divided by C squared. Now, in this case, then if we go ahead just like before trying to solve for V, then that means one minus V squared divided by C squared is going to be equal to 0.98 squared, which means V squared divided by C squared equals one minus 0.98 squared. OK? Which means then that V squared equals C squared multiplied by that expression. OK? And no V is going to be equal to C multiplied by the square root of one minus 0.98 squared. And when we go ahead and calculate, we get that value to be equal to 0.198997 C. And if we rounded to three significant figures, it equals approximately 0.199 C does for a 2% for a 2% slower. T the speed is going to be approximately 0.197 C and for a 2% contraction in length, the speed is 0.199 C. If we look at back at our answer choices that tells us D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

At what speed do the relativistic formulas for (a) length and (b) time intervals differ from classical values by 1.00%? (This is a reasonable way to estimate when to use relativistic calculations rather than classical.)

Key Concepts

Relativity

Lorentz Transformation

Classical vs. Relativistic Physics

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