2. Einstein's theory of relativity states that the energy E of an object is related to its velocity v (m: mass of the object, c: speed of light) in the following way: mc? E = In this problem, we will write out a Taylor polynomial describing the energy and try to understand when relativistic behaviors start to play a role. (a) Write down the Taylor series for E in terms of v. (b) We need to understand what we just did: For which v is the formula valid? Would those v, for which the formula is not valid, even make physical sense in this example? (c) Write out explicitly the Taylor polynomial of degree 4 for E. This should contain three terms: The first two should be familiar to you. What do they describe? To answer this, you need to use both the mathematical meaning of each term as well as your physical knowledge. (d) The third term in the degree 4 polynomial is the first relativistic correction to the classical kinetic energy. All further terms in the Taylor series are relativistic corrections to the classical kinetic energy. This raises the question: When does such a correction need to be made? What is the magnitude of the error being made by ignoring relativistic effects in classical physics? To answer this we will study the error Rf () made by considering only the Taylor polynomial of degree 2 for E and not using any of the relativistic terms. i. Use the error from Pr. 1 to show that the the error Rf () satisfies 3m 1. |Rf 5/2 2 12 ii. Using e = 3- 10$m/s, calculate the exact error made in ignoring relativistic contributions for a human (100 kg) moving at 100 m/s. What is the difference (in magnitude) between the error you calculated and the classical kinetic energy (mv²) of this person?
2. Einstein's theory of relativity states that the energy E of an object is related to its velocity v (m: mass of the object, c: speed of light) in the following way: mc? E = In this problem, we will write out a Taylor polynomial describing the energy and try to understand when relativistic behaviors start to play a role. (a) Write down the Taylor series for E in terms of v. (b) We need to understand what we just did: For which v is the formula valid? Would those v, for which the formula is not valid, even make physical sense in this example? (c) Write out explicitly the Taylor polynomial of degree 4 for E. This should contain three terms: The first two should be familiar to you. What do they describe? To answer this, you need to use both the mathematical meaning of each term as well as your physical knowledge. (d) The third term in the degree 4 polynomial is the first relativistic correction to the classical kinetic energy. All further terms in the Taylor series are relativistic corrections to the classical kinetic energy. This raises the question: When does such a correction need to be made? What is the magnitude of the error being made by ignoring relativistic effects in classical physics? To answer this we will study the error Rf () made by considering only the Taylor polynomial of degree 2 for E and not using any of the relativistic terms. i. Use the error from Pr. 1 to show that the the error Rf () satisfies 3m 1. |Rf 5/2 2 12 ii. Using e = 3- 10$m/s, calculate the exact error made in ignoring relativistic contributions for a human (100 kg) moving at 100 m/s. What is the difference (in magnitude) between the error you calculated and the classical kinetic energy (mv²) of this person?
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I need a b and c answered
Transcribed Image Text:2. Einstein's theory of relativity states that the energy E of an object is related to its velocity v (m:
mass of the object, c: speed of light) in the following way:
mc?
E =
In this problem, we will write out a Taylor polynomial describing the energy and try to understand
when relativistic behaviors start to play a role.
(a) Write down the Taylor series for E in terms of v.
(b) We need to understand what we just did: For which v is the formula valid? Would those v, for
which the formula is not valid, even make physical sense in this example?
(c) Write out explicitly the Taylor polynomial of degree 4 for E. This should contain three terms:
The first two should be familiar to you. What do they describe? To answer this, you need to use
both the mathematical meaning of each term as well as your physical knowledge.
(d) The third term in the degree 4 polynomial is the first relativistic correction to the classical kinetic
energy. All further terms in the Taylor series are relativistic corrections to the classical kinetic
energy. This raises the question: When does such a correction need to be made? What is the
magnitude of the error being made by ignoring relativistic effects in classical physics? To answer
this we will study the error Rf () made by considering only the Taylor polynomial of degree
2 for E and not using any of the relativistic terms.
i. Use the error from Pr. 1 to show that the the error Rf () satisfies
3m
1.
|Rf
5/2 2
12
ii. Using e = 3- 10$m/s, calculate the exact error made in ignoring relativistic contributions for
a human (100 kg) moving at 100 m/s. What is the difference (in magnitude) between the
error you calculated and the classical kinetic energy (mv²) of this person?
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