Assignment 3

Problem 1:

A sonar device emits a sound with a frequency of 30 kHz as it approaches a wall and measures the frequency of the echo as 30.9 kHz.
If the speed of sound is 340 m/s, what is the speed with which the device approaches the wall?

Solution:

Problem 2:

Two point sources emit spherical waves with wave number k and frequency ω into 3D space.  One source is located at the origin and one source a small distance r away from the origin.   A detector is located far away at R, with R >> r.
For each source, let Ψi = A exp(i(kri - ωt + φi))/ri, where ri is the distance of the detector from the source and A is real.  For the source at the origin choose φ = 0, while for the source at r choose a non-zero φ.
(a)  Write down an expression |Ψtotal|2 (proportional to the average intensity) at the detector.  Make the far-field approximation, i.e. lines from each source to the detector are parallel to each other.
(b)  Let k = kR/R.  What condition must hold for k·r, so that the average intensity at the detector is maximized?
(c)  What happens if another point source is added at position 2r with phase 2φ.
(d)  For case (c), what condition must hold for k·r, so that the average intensity at the detector zero?

Solution:

Problem 3:

It is known that if a point-like source light is placed in the focus F of a concave parabolic mirror the reflected rays are parallel to the optical axis.  Suppose the parabolic mirror is formed by the rotation of a parabola y = Ax2 around the y axes.  Find the focal distance of the concave parabolic mirror.

Solution:

Problem 4:

A laser beam is incident at an angle of 30.0o to the vertical onto a solution of corn syrup in water.  If the beam is refracted to 19.24o to the vertical,
(a)  What is the index of refraction of the syrup solution?
Suppose that the light is red, with vacuum wavelength 632.8 nm.  Find its
(b)  wavelength,
(c)  frequency, and
(d)  speed in the solution.
Give numerical answers.

Solution:

Problem 5:

238Pu decays by α-emission with a half-life of 87 years.
238Pu --> 234U + α.
The half-life of 234U is much longer, 3.5*105 years (ignore this decay).
The heat produced in this decay can be converted into useful electricity by radio-thermal generators (RTG's).  The Voyager 2 space probe, which was launched in August 1977, flew past four planets, including Saturn, which it reached in August 1981.
How much plutonium would an RTG on Voyager 2 with 5.5 % efficiency have to carry at the start to deliver at least 395 W of electric power when the probe flies past Saturn?

Masses:
4
He:  4.002603 u,
234
U: 234.040947 u
238
Pu:  238.049555 u
Conversion:  1 u = 931.5 MeV

Solution: