Section A
Q1.
(a) Consider a particle of mass m moving in a potential field of the form V(\vecr) = V0(x2 + y2), where V0 is a constant. What are the conserved physical quantities for the particle?
(b)
Two spaceships are approaching each other as depicted below.
Spaceship 1 is moving with a velocity \overrightarrowu1 = 0.7c \hatx, while spaceship 2 is moving with a velocity \overrightarrowu2 = -0.8c \hatx as measured by the stationary observer on Earth. Here, c is the velocity of light in vacuum and \hatx is the unit vector along the x-axis.
(i) What, according to Galilean addition of velocity, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c)?
(ii) What, according to relativistic addition of velocities, is the speed of spaceship 2 with respect to spaceship 1 (in terms of c)?
(iii) Spaceship 1 emits a light signal towards spaceship 2. The observer on spaceship 2 measures the frequency of the light signal to be 1016 Hz. What was the original frequency of the light signal?
(c) A tuning fork A produces 6 beats per second with another tuning fork B of frequency 508 Hz. When the prongs of A are loaded with a little wax, number of beats heard are 4 per second. Find the frequency of A before it was loaded with wax.
(d) A transmission hologram is recorded in a medium of refractive index 1.15. If the spacing between the two consecutive maxima is 557 nm and the angle of incidence of reference and object waves from the reference axis (say z-direction) is 30°, find the wavelength of the light source.
(e) The coherence length and the bandwidth of a light source is 3 km and 1 × 10-5 nm, respectively. Determine its wavelength and the pulse duration.
