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Answer: Internal forces do not perform any work in the virtual displacement. This is not based on Newton's III law. In fact, the III law is a result of this rather than the cause. As you read the equation on slide 34 from left to right, you will find that we have not employed in that relation that action and reaction are equal and opposite. As the argument continues on slide 35, you will find that for the result on the first equation of slide 35 to hold for an *ARBITRARY* displacement (and this is the key issue), it follows that the sum of all the internal forces vanishes, which is the conclusion reached at the bottom of slide 35 - the conclusion being that the total momentum is independent of time. Newton's III law then follows from this as we find on slide 36.
Answer: Please let me know exactly which slide number in which lecture is referred to in your question. I suspect that you are concerned about energy conservation. While the context of your question is not clear to me, I think the answer lies in recognizing that reflection of light accounts only for part of the incident energy. For example, some energy gets transmitted/refracted. Total energy is, of course, conserved and this is easily established when you take account of the complete physical process.
Answer: The central issue in this discussion is to first understand just what the 'paradox' is, and then resolve it within the framework of the STR. The main 'paradox', as it would seem, is the asymmetry between the perceptions of the twins, as pointed out on Slide no.58 at the following link: http://www.nptel.iitm.ac.in/courses/115106068/STiCM/Unit%2006%20L19%20to%20L22%20-%20Special%20Theory%20of%20Relativity.pdf The perceptions of equivalent observers would differ if the non-relativistic addition formula were used. The velocity addition formula has been used to resolve this paradox. It shows that the conclusions of the two observers in inertial frames do not differ, thereby resolving the 'paradox'.
Answer: The equivalence principle (Slide 59 of Unit 06) requires the conclusions of the two observers in relative motion with respect to each other to be the same. The paradox concerns only two observers in relative motion with respect to each other, which means that one of the observers must be in the rest-frame and the other moving with respect to the first. S' moves at velocity v with respect to S, which means that S is assumed to be at rest (home-bound observer's frame). In your question you seem to refer to three - not two - frames: a rest frame with respect to which two other frames are moving at velocities v and v' respectively. The twin-paradox is concerned with the equivalence principle mentioned above, and not to the situation you have referred to.
Answer: The special theory of relativity emphasizes the fact that laws of physics are the same in all inertial frames of reference. While holding this, STR reconciles with the fact that the speed of light is invariant and independent of the state of motion of the observer, whether one of rest, or at constant velocity with respect to the observer at rest. A natural consequence of this is that neither the euclidean 3-dimensional space interval nor the orthodox time interval is invariant under Lorentz Transformation (LT). Now, the velocity is a ratio of distance over time, and neither the numerator nor the denominator in this interval is independently invariant under LT. However, if one now constructs/defines "proper velocity" as the ratio of "proper length" divided by "proper velocity" as how we have defined these, then one can extend the idea of "proper velocity" (which has 3 components as defined above) to include a 4th component giving a velocity 4-vector which is Lorentz invariant.
Answer: Length contraction and time dilation are terms that connect length and time intervals in two frames of references moving with respect to each other. This motion is relative, so one must fix one frame (S) as fixed and the other (S') moving w.r.t. S. Once this is done, there is no ambiguity between contraction/expansion. The equivalence of the two frames is best illustrated in the resolution of the twin-paradox (Lecture 21).
