# Swimming Pool Problem

## Homework Statement

You and a friend are at opposite sides of a large swimming pool, 20.0 m apart each holding onto a large inflated ball. You decide to launch water waves at each other by moving the balls downward into the water and back up at a tae of one cycle per second. You both push the balls downward at the same time each cycle perfectly synchronised. Your action launches surface waves along the water. In reality the wave profile will be distorted where you and you friends are but you may assume that the waves can be described as perfectly sinusoidal functions of position and time.

(a) What is the frequency f, in units of Hz, of the waves that you and your friend generate?
(b) The water wave speed V is 10.0 m/s. What is the wavelength λ of the waves that you and your friend generate?
(c) Along the line between you and your friend, in the path of the waves that you are generating, there is a third ball on the surface of the water. You expect to see the ball moving up and down under the force of the waves, but to your surprise it sits exactly still. Answer the following two questions. (You should be able to do this without any calculations by sketching the water wave profile along the line between you and your friend.)
(i) How many locations are there along the line between you and your friend where this could happen?
(ii) How far apart are the different locations?
(d) Now you and your friend stand at the same side of the pool, 20.0 m apart, and you both launch waves the same way as before, at same frequency as before
and perfectly synchronized. The waves radiating outward from the two of you interfere constructively at an angle of 0.00° (defined as the normal to your side of the pool). What is the next angle at which the waves interfere constructively? Report your answer to three significant figures.
(e)Electromagnetic (light) waves are composed of individual quanta of energy (photons), and there is a simple relationship between the energy E and the wave frequency ν. The exact same relationship holds for water and sound waves. State the relationship and use it to calculate the energy in a single quantum ("phonon") of the water wave.
(f) Individual photons can dislodge individual electrons from metals (the photoelectric effect) if the photon energy exceeds the binding energy between the metal and the electron (i.e. the work function). You notice that your friend, who had slipped in the mud before entering the pool, is gradually getting cleaner due to the action of the waves. You suspect that individual quanta of the water waves are dislodging individual dirt particles, which come off of your friend with a particle speed that you estimate to be 1.0 x 10-3 m/s.
(i) Based on your result from part (e), and assuming a negligibly small binding energy between the dirt particles and your friend, calculate the mass of a particle that could be dislodged with the speed you estimated by this quantum mechanical washing mechanism.
(ii) The dirt particles that you see coming off of your friend are at least 1 mg in mass. Does the quantum mechanical washing mechanism explain how the water is washing your friend, or is the washing taking place through classical mechanical processes?

2. The attempt at a solution

(a) I think f= 1 Hz
(b) λ= 10 m
(c) (i) 4 locations
(ii) The different locations are λ/2 m apart

Mentor

d) Do you have a sketch of the setup? It will help you a lot.
If you know how a double-slit experiment works: It is the same setup here.
e) ##E=h\nu##?
f) needs e) and the formula for the kinetic energy of moving objects.

I do not have a sketch of the setup. It is only given as a text

for f) the formula is 1/2mv^2=hf-[work function]

Mentor
I do not have a sketch of the setup. It is only given as a text
You should make one yourself ;).

For f, you can neglect the binding energy ("work function").

Ok. Is the sketch similar to the one I used in order to answer question c)(i)
If so, how to I use it for question d) ?

Mentor
Probably not, it has to be two-dimensional - something similar to this, where the slits are replaced by the balls.

I still don't understand how to proceed..

Just want to verify question e:

E=hf=6.626*10^-34*1=7*10^-34

Question f(i): 1/2mv^2=hf

Which means that: m=2hf/v^2
Therefore: m=(2*7*10^-34)/(1.0*10^-3)=1.4*10^-30 g

Mentor
You should use units everywhere (and be careful with g<->kg), and square the velocity, but the concept looks good.

Concerning d: You can look for explanations of the double-slit experiment and try to understand them.