Question

In: Physics

1. Suppose that Alice is rushing past Bob at a velocity u, and she carries with...

1. Suppose that Alice is rushing past Bob at a velocity u, and she carries with her a block of transparent material with index of refraction n. If she shines light through that material then she sees the light moving through it with a velocity c/n. What speed does Bob see the light move through that material? Check that your result has the correct limiting behavior that when n = 1, and u << c/n!

2. An unstable particle of mass M sits at rest in the laboratory. It decays into two identical particles each of mass m which go flying out in opposite directions. What is the velocity of the decay particles, as measured in the lab frame? Hint: you can check that you get the right limit when the mass m goes to zero!

Solutions

Expert Solution

(1)

Speed of light in a transparent material of refractive index n in Alice frame is

Alice is moving relative to Bob at speed u.

Using Einstein's velocity addition rule, the velocity of light in the transparent material in Bob's frame is

Using v=c/n

For n=1, the speed of light in vacuum in Alice frame is c. We will check whether the speed of light in vaccum is c in Bob's frame also. Substituting n=1 in above relation, we get

The speed is equal to the speed of light in Bob's frame. Our relation behave correctly in this limit.

For u << c/n, when the speed of Bob's frame is very small relative to the speed of light in the material, u can be neglected when addition to c. This is the non-relativistic case, we expect the speed of light in Bob's frame to be c/n.

When Alice is moving at non-relativistic speed, the speed of light in the transparent medium is equal in both frames.

(2)

Initially mass M is at rest, the total energy of the system is equal to the rest mass energy of the particle

The velocity of the daughter particles is equal in magnitude and opposite in direction. Let v₁=v₂=v be the speed of the two daughter particles. The total energy of the daughter particles is

By the conservation energy

When the two identical particles are photons. We know that mass of the photons is zero. Substituting m=0

The speed of the daughter particles is c. We know that photons travel at speed c, therefore our relation gives correct result in limit m goes to zero.

genius_generous answered 2 years ago

Next > < Previous

Related Solutions

Suppose Alice flips 4 coins and Bob flips 4 coins. Find the probability that Alice and...

Suppose Alice flips 4 coins and Bob flips 4 coins. Find the probability that Alice and Bob get the exact same number of heads.

How to generate a key pair for Alice and Bob Respectively Suppose Alice sends plaintext P=...

How to generate a key pair for Alice and Bob Respectively Suppose Alice sends plaintext P= 113, how does she encrypt and whats the ciphertext C? After Bob receives C, how does he decrypts it to get the plaintext P? Suppose Alice sends plaintext P= 113, how does she sign it and what are sent to Bob. How does Bob verify the signature? Suppose Bob sends plaintext P=113, how does he sign it and what are sent Alice. How does...

RSA: Alice wishes to send Bob the message POET. Suppose Bob chooses P = 29, Q...

RSA: Alice wishes to send Bob the message POET. Suppose Bob chooses P = 29, Q = 31, E = 47, and D = 143. Show the steps that Alice uses to encrypt the message POET (use the ascii values of the letters P, O, E, and T), and how Bob decrypts the message he receives from Alice. You will be generating very large numbers, and will find the following calculator helpful: https://www.calculator.net/big-number-calculator.html

Suppose Alice and Bob have RSA public keys in a file on a server. They communicate...

Suppose Alice and Bob have RSA public keys in a file on a server. They communicate regularly, using authenticated, confidential message. Eve wants to read the messages but is unable to crack the RSA private keys of Alice and Bob. However, she is able to break into the server and alter the file containing Alice’s and Bob’s public keys. (1) How should Eve alter the file to so that she can read confidential messages sent between Alice and Bob, and...

Problem 4 | A modied man-in-the-middle attack on Diffie-Hellman Suppose Alice and Bob wish to generate...

Problem 4 | A modied man-in-the-middle attack on Diffie-Hellman Suppose Alice and Bob wish to generate a shared cryptographic key using the Diffie-Hellman protocol. As usual, they agree on a large prime p and a primitive root g of p. Suppose also that p = mq + 1 where q is prime and m is very small (so p - 1 = mq has a large prime factor, as is generally required). Since g and p are public, it is...

Question 1 - Game Theory (7 Marks) Alice and Bob are playing a game of simultaneous...

Question 1 - Game Theory Alice and Bob are playing a game of simultaneous moves where they are deciding which concert they will attend on Saturday night. Two musicians are performing that night locally, Artist 1 and Artist 2. Alice prefers to go to Artist 1 with Bob rather than Artist 2 with Bob, and she prefers going to Artist 2 with Bob rather than going to Artist 2 alone. Moreover, she prefers going to Artist 1 alone rather than...

1. Bob wants to use DES to make a secured communication with Alice using DES key...

1. Bob wants to use DES to make a secured communication with Alice using DES key ‘Kb’ to encrypt and decrypt data. Bob uses PKI method to distribute the DES key to Alice. If Alice private key is ‘Kpa’, and her public key is ‘Kpb’. Please describe the process that Bob can send his DES key ‘Kb’ to Alice securely. 2. You receive a message that was encoded using a block encoding scheme with the encoding matrix M=3275 Verify by...

At 9/1/2021 the capital balances are Alice 88870 Bob 68327 Carol 36294 David joins the partnership...

At 9/1/2021 the capital balances are Alice 88870 Bob 68327 Carol 36294 David joins the partnership by contributing $28687 for 10% of the capital. Under the bonus method, how much capital does Alice gain as a bonus? Answer: 2911 Under the goodwill method, what is Bob's capital position after David joins?

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ =...

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉. Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.

Suppose U is uniform on (0,1). Let Y = U(1 − U). (a) Find P(Y >...

Suppose U is uniform on (0,1). Let Y = U(1 − U). (a) Find P(Y > y) for 0 < y < 1/4. (b) differentiate to get the density function of Y . (c) Find an increasing function g(u) so that g(U ) has the same distribution as U (1 − U ).

Subjects