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Jethin P.
Was teaching assistant for four semesters in WSU Pullman, Majored in Physics
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Calculus
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Question:

Given that a function satisfies the following: f(x+y) = f(x) f(y) and f'(0) = ln(4). Find the function(s) f(x) ?

Jethin P.
Answer:

Clearly, exponential function satisfy the condition f(x+y) = f(x) f(y), i.e., for f(x) = z^{ a x } we have, z^{ a (x + y) } = z^{ a x } * z^{ a y } ------------------- (1) where, a is constant. f(1) = 4 , implies z^a = 4. Now, d/dx f(x) = d/dx z^{ a x } = a * z^{ a x } * ln(z), Therefore, f'(0) = a ln(z) = ln( z^a) = ln(4). Thus solutions for z and a are 1) z =4 and a = 1 or 2) z = 2 and a = 2 giving two solutions for f(x) namely, 1) f(x) = 4^x 2) f(x) = 2^{2*x}

Advanced Physics (Special Relativity)
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Question:

Does the mass of an object vary depending on the velocity of its motion? More precisely, whether the mass of an object measured in a rest frame (The frame where you are at rest) and in a moving frame differ? Explain.

Jethin P.
Answer:

This question can be answered very concretely using group theory concepts. Let me first explain the relevant group theory concepts. One has to use special theory of relativity in order to explain the kinematics and dynamics of the particles in different inertial frames. The underlying lie group of special relativity is Poincare group. Each element of this group represent different inertial frames. For every lie group, one can construct special operators called casimir operators. These operators are invariant under the transformations (basically an element of the group) of the associated lie group. For Poincare group, there are two casimir operators that one can construct, namely, P^{\mu} P_{\mu} and W^{\mu} W_{\mu}, where P^{\mu} and W^{\mu} is four-vector momentum and Pauli-Lubanski four-pseudovector, of the particle respectively. The corresponding invariant eigenvalue of these operators are "m^2" and "-m^2 * s(s+1)", respectively, where "m" is the mass and "s" is the spin of the particle. What really happens is when the velocity of the particle is big enough to consider it relativistically, The linear relation between the momentum of the particle and its velocity no longer hold true. In fact, The relation between the momentum of the particle and its velocity becomes non-linear at relativistic speeds.

Physics
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Question:

Why is Newton's law of gravity inaccurate?

Jethin P.
Answer:

According to Einstein's special theory of relativity no information can go faster than the speed of the light. Now assume that our Sun suddenly vanishes from where it is. From the Newton's law of gravity (There's no time dependence in the law), one can conclude that the effect of Sun being vanished should be simultaneously experienced by the Earth, and our planet should just stop revolving around Sun and then continued moving along a straight line. In reality this cannot happen as the information of Sun being vanished reaches the Earth only after roughly eight minutes later. Till that time the Earth will continue revolving around the point where the Sun originally was. To explain this Einstein developed theory called General relativity.

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