Question

Classical mechanics is an extremely well tested model. Hundreds of years worth of experiments, as well as most feats of engineering, have verified its validity. If special relativity gave very different predictions than classical physics in everyday situations, it would be directly contradicted by this mountain of evidence. In this problem, you will see how some of the usual laws of classical mechanics can be obtained from special relativity by simply assuming that the speeds involved are small compared to the speed of light.Two of the most surprising results of special relativity are time dilation and length contraction, namely, that measured intervals in time and space are not absolute quantities but instead appear differently to different observers. The equations for time dilation and length contraction can be written t=?t0 and l=l0/?, where?=11?u2c2?.Part AFind the first two terms of the binomial expansion for ?.Express your answer in terms of u and c.Hints? = 1+12(uc)2 … SubmitMy AnswersGive UpCorrectYou can see that ??1 if u?c, as is the case in most situations. If you set ?=1 in the equations for time dilation and length contraction you recover the equations of classical physics, which state essentially that there is no time dilation or length contraction. Therefore, we don't see any appreciable length contraction or time dilation in everyday life.Part BConsider a case involving a speed that is fast compared to those encountered in our everyday life: a spy plane moving at 1500m/s. Find the deviation from classical physics (??1) that relativity predicts at this speed. Use only the first two terms of the binomial expansion, as your calculator may not be able to handle the necessary number of digits otherwise.Express your answer to four significant figures.??1 = 1.250×10?11SubmitMy AnswersGive UpCorrectIf you lived for 70 years in such a spy plane moving at 1500m/s, this would amount to about 28ms of cumulative time difference between you and people who lived at rest relative to the earth when you finally landed. Thus, it is not surprising that relativistic effects are not observed in everyday life, or even at the fringes of everyday life. By using atomic clocks, which can measure time accurately to one part in 1013 or better, the time dilation at the normal speed for an airliner has been verified.Part CNow, consider the relativistic velocity addition formula:speed=v+u1+vuc2.If v=u=0.01c=1% of c, what is the relativistic sum of the two speeds?Express your answer as a percentage of the speed of light to five significant figures.

Answer 1

Classical mechanics is a well-tested model that can be derived from special relativity in everyday situations where speeds are small compared to the speed of light. Time dilation and length contraction are two surprising outcomes of special relativity that are not observed in everyday life. The deviation from classical physics predicted by relativity can be calculated for specific speeds and can result in small cumulative time differences.

Step-by-step explanation:

Classical mechanics is a well-tested model that has been verified by hundreds of years worth of experiments and engineering feats. However, special relativity can still be derived from classical mechanics in situations where speeds are small compared to the speed of light. Two surprising outcomes of special relativity are time dilation and length contraction, which show that measured intervals in time and space are not absolute quantities and can appear differently to different observers.

The equations for time dilation and length contraction are t = t0/sqrt(1 - (u^2/c^2)) and l = l0/sqrt(1 - (u^2/c^2)), respectively, where u is the speed of the object and c is the speed of light. When u is much smaller than c, these equations reduce to the equations of classical physics, meaning that time dilation and length contraction are not observed in everyday life.

In the case of a spy plane moving at 1500m/s, which is fast compared to everyday situations, the deviation from classical physics predicted by relativity (??1) can be calculated using the binomial expansion and the first two terms. The result is approximately 1.250×10^-11, meaning there would be a small cumulative time difference of about 28ms after living for 70 years in such a spy plane compared to people at rest relative to the Earth.

Finally, the relativistic velocity addition formula states that the sum of two speeds in relativity is given by speed = (v + u)/(1 + (vu/c^2)). In the case of v = u = 0.01c, the relativistic sum of the two speeds is approximately 0.02010c, or 2.010% of the speed of light.

Answer 2

The Answer is 0.019998c

Step-by-step explanation:

Please see the attached Picture for the answer.