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*Answer can only contain numeric values

QUESTION: 1

An observer O at rest midway between 2 sources of light at x = 0 and x = 10m, observes the 2 sources to flash simultaneously. According to a second observer O', moving at a constant speed parallel to the x-axis, one source flashes 13ns before the other. Find the speed of O' relatives to O (in units of c)

Solution:

A → Flashing observed from first source (at ** x** = 0)

B → Flashing observed from second source (at ** x** = 10)

Now Event (*A*) occurs (at *x* = 0) Event ** B** occurs (at

For *O * *t* = 0 *t* = 0

For* O'*

Here we assumed that both observers observe event ** A** at

Now the problem says that for ** O'**, one event occurs 13

∴ O' observes event

Now, applying transformation for time

The correct answer is: 0.363

*Answer can only contain numeric values

QUESTION: 2

What is the speed of a particle having a momentum of 5MeV/c and a total relativistic energy of 10 MeV (in terms of *c*)?

Solution:

The correct answer is: 0.5

*Answer can only contain numeric values

QUESTION: 3

A tube of water is travelling at c/2 relative to the lab frame when a beam of light travelling in the same direction as the tube enters it. What is the speed of light in the water relative to the lab frame? (The index of water is 4/3)

Solution:

The beam of light travels through the water at a speed while the tube itself is travelling at a speed c/2.

Using the addition of relativistic velocities

= 0.909 *c*

The correct answer is: 0.909

*Answer can only contain numeric values

QUESTION: 4

A particle of mass M decays from rest into 2 particles. One particle has mass m and the other particle is massless. Find the momentum of the massless particle (in units of

Solution:

Momentum conservation

** P_{initial}** = 0

*mv* + * p *= 0

Now, for the masses particle ** E = p**. For the particle of mass

from conservation of energy, [Take *c* = 1]

∴ momentum of masses particle is

The correct answer is: 0.5

*Answer can only contain numeric values

QUESTION: 5

The Lyman α spectral line of hydrogen (λ = 122nm) differs by 1.8 × 10^{–12} m in spectra taken at the opposite ends of the sun’s equator. What is the speed of a particle on the equator due to the sun’s rotation, (in km/s)

Solution:

When the sun rotates, one end of the equator moves towards us & the other moves away from us.

So, the difference given is basically the difference between the red & blue shift

Now, using the non relativisitic doppler effect (since the speed of sun is much less than ** C**)

** v** = 2.2

The correct answer is: 2.2

*Answer can only contain numeric values

QUESTION: 6

A distant galaxy is observed to have its hydrogen β line shifted to a wavelength of 580*nm*, away from the lab value of 434*nm.* Find the approximate velocity of the recession of the distant galaxy (in terms of *c*)

Solution:

Original wavelength = 434 *nm*

shifted λ = 580 nm

(given in the problem)

Relativistic dopper effect formulae.

= 0.28

The correct answer is: 0.28

*Answer can only contain numeric values

QUESTION: 7

Two space ships approach Earth with equal speeds as measured by an observer on Earth, but from opposite directions. A meter stick on one spaceship is measured to be 60 *cm* long by an occupant of the other ship. What is the speed of each spaceship as measured from earth. (in units of *c*)?

Solution:

length of stick on ship 1 measured by ship 2 = 60 *cm*

Now, since each ship is moving with velocity *v*.

∴ Velocity of ship 2 wrt ship 1 = 2*v*

*l*_{0}* *= 1** m** (meter stick)

Speed of each ship = 0.4

The correct answer is: 0.4

*Answer can only contain numeric values

QUESTION: 8

In inertial frame S_{1} two events occur at the same time and 3c minutes aparts in space. In the inertial frame S', the same events occur at 5c minutes apart. What is the time interval between the events in S' ? (in mins)

Solution:

In frame

In frame

Since is an invariant quantity

= 25*c*^{2} – 9*c*^{2}

= 16*c*^{2}

Δt_{2} = 4c In frame* S'*

The correct answer is: 4

*Answer can only contain numeric values

QUESTION: 9

A photon strikes an electron of mass m that is initially at rest, creating an electron positron pair. The photon is destroyed and the positron and 2 electrons move off at equal speeds along the initial direction of the photon. Find energy of the photon (in units of mc^{2}).

Solution:

Let us work in the system of units in which ** c** = 1

Now since ** E^{2} – p^{2}** is an invariant quantity, we equate this in the initial laboratory frame and in the final frame of particles

In lab frame, the electron is at rest.

∴ ** E_{electron}** =

∴ Total Energy of the system is ** E + m** = Energy of photon + Energy of electron

∴ Laboratory frame ...(1)

In the final state of particles

*E*^{2} – *p*^{2} = (*m* + *m* + *m*)^{2}

= (3** m**)

= 9

Equating for the 2 frames (1) = (2)

∴ (** E + m**)

since a photon is massless

∴ ** E** =

∴

2

Now in SI units

The correct answer is: 4

*Answer can only contain numeric values

QUESTION: 10

The half life of a π^{+} meson at rest is 2.5 × 10^{–8} s. A beam of π^{+} mesons is generated at a point 15m from a detector. Only half of the π^{+} mesons live to reach the detector. What is the speed of the π^{+} mesons? (in terms of c)

Solution:

In the rest frame of mesons,

Life time (in rest frame of mesons) = 2.5 × 10^{–8} *s
*

= 0.89

The correct answer is: 0.89

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