Archive for the Cute Problems Category

The Problem of the Disintegrating Asteroid

Posted in Cute Problems, The Universe and Stuff with tags , , on September 30, 2019 by telescoper

I thought you might enjoy this entry in the Cute Problems folder.

An asteroid is moving on a circular orbit around the Sun with an orbital radius of 3AU when it spontaneously splits into two fragments, which initially move apart along the direction of the original orbit. One fragment has a speed which is a fraction 0.65 of the original speed, the other has a speed of 1.35 times the original speed. The original orbit (solid line) is shown above, along with the two new orbits (dashed and dotted).

  1. Which orbit does the fast fragment follow, and which the slow fragment?
  2.  Calculate the original orbital speed in AU/year.
  3. Calculate the angular momentum per unit mass, h, of the original asteroid and of each of the two fragments in units of AU2 per year. [HINT: Show that in these units, for a general orbit of eccentricity e and semi-major axis a, h2=4π2 a (1-e2).]
  4.  Calculate the eccentricities of the orbits of the two fragments.
  5.  Calculate the orbital periods of the two fragments in years.

Answers please through the Comments box. First complete set of answers wins a trip to the Moon on gossamer wings.

 

 

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A Problem of Dimensions

Posted in Cute Problems, Maynooth, The Universe and Stuff with tags , , , on August 21, 2019 by telescoper

We’ve more-or-less sorted out who will be teaching what next term in the Department of Theoretical Physics at Maynooth University next term (starting a month from now) and I’ll be taking over the Mathematical Physics module MP110, which is basically about Mechanics with a bit of of special relativity thrown in for fun. Being in the first semester of the first year, these is the first module in Theoretical Physics students get to take here at Maynooth so it’s quite a responsibility but I’m very much looking forward to it.

I am planning to start the lectures with some things about units and dimensional analysis. Thinking about this reminded me that I posted a dimensional analysis problem (too hard for first-year students) on here a while ago which seemed to pose a challenge so I thought I would post another here for your amusement.

 

The period P for an elliptical orbit of semi-major axis a of  a moon of mass m around a planet of mass M, depends only on the quantities  a, m, M and G (Newton’s Gravitational Constant).

(a). Using dimensional analysis only, determine as completely as possible the relationship between P and these four quantities.

(b). How would the period P compare with the period P′ of a system consisting of a moon of mass 2m orbiting a planet of mass 2M in an ellipse with the same semi-major axis a?

Please submit your efforts through the comments box below.

 

More Order-of-Magnitude Physics

Posted in Cute Problems with tags , , , on April 25, 2019 by telescoper

A very busy day today so I thought I’d just do a quick post to give you a chance to test your brains with some more order-of-magnitude physics problems. I like using these in classes because they get people thinking about the physics behind problems without getting too bogged down in or turned off by complicated mathematics. If there’s any information missing that you need to solve the problem, make an order-of-magnitude estimate!

Give  order of magnitude answers to the following questions:

  1. What is the maximum distance at which it could be possible for a car’s headlights to be resolved by the human eye?
  2. How much would a pendulum clock gain or lose (say which) in a week if moved from a warm room into a cold basement?
  3. What area would be needed for a terrestrial solar power station capable of producing 1GW of power?
  4. What mass of cold water could be brought to the boil using the energy dissipated when a motor car is brought to rest from 100 km/h?
  5. How many visible photons are emitted by a 100W light bulb during its lifetime?

There’s no prize involved, but feel free to post answers through the comments box. It would be helpful if you explained a  bit about how you arrived at your answer!

A Problem of Sons

Posted in Cute Problems with tags , , on January 31, 2019 by telescoper

I’m posting this in the Cute Problems folder, but I’m mainly putting it up here as a sort of experiment. This little puzzle was posted on Twitter by someone I follow and it got a huge number of responses (>25,000). I was fascinated by the replies, and I’m really interested to see whether the distribution of responses from readers of this blog is different.

Anyway, here it is, exactly as posted on Twitter:

Assume there is a 50:50 chance of any child being male or female.

Now assume four generations, all other things being equal.

What are the odds of a son being a son of a son of a son?

Please choose an answer from those below:

 

UPDATE: The answer is below:

 

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Acute Geometry Problem

Posted in Cute Problems, mathematics on December 3, 2018 by telescoper

I saw a plea for help on Twitter from Astronomer Bryan Gaensler who is stuck with his son’s homework.

So please give him a hand by solving this to find a, b and c.

Your time starts now.

Hubble Problems

Posted in Cute Problems, The Universe and Stuff with tags , on October 12, 2018 by telescoper

Here I am, only connecting again.

Almost every day I get a spam message from a certain person who thinks he can determine the Hubble constant from first principles using  biblical references. The preceding link takes you to an ebook. I was thinking of buying it, but at 99c* I considered it prohibitively expensive.

*I am informed that it has now gone up to £1.30.

My correspondent also alleges that in writing this blog I am doing the Devil’s work. That may be the case, of course, but I can’t help thinking that there must be more effective ways for him to get his work done. Either that or he’s remarkably unambitious.

Anyway, to satisfy my correspondent here is one for the problems folder:

Using  the information provided in Isaiah Chapter 40 verse 22, show that the value of the Hubble constant is precisely 70.98047 km s-1 Mpc-1.

You may quote the relevant biblical verse without proof. In the King James version it reads:

40.22. It is he that sitteth upon the circle of the earth, and the inhabitants thereof are as grasshoppers; that stretcheth out the heavens as a curtain, and spreadeth them out as a tent to dwell in.

By the way, please note that the inverse of the Hubble constant has dimensions of time, not distance.

Answers into my spam folder please (via the comments box).

 

While I am on the subject of Hubble, I will mention the news that the Hubble Space Telescope is having a few technical problems as a result of a failure of one of its gyros. In fact a few days ago it went into `safe mode’ to help engineers diagnose and fix the problem, during which time no observations are being taken. I’m told by people who know about such things that the spacecraft can actually operate on only one gyro if necessary, using information from other systems for attitude control, so this problem is not going to be terminal, but it will slow down the pointing quite a bit thus make it less efficient. With a bit of luck HST will be back in operation soon.

 

 

 

 

A Problem with a Geostationary Orbit

Posted in Cute Problems, The Universe and Stuff with tags , , , on September 26, 2018 by telescoper

I’ve been sorting through some old problem sets for my course on Astrophysics and Cosmology, and thought I would post this one in the Cute Problems folder for your amusement. The first part is easy, the second part not so much…

  1. Verify that the radius of a circular geostationary orbit around the Earth is about 42,000 km, i.e. find the radius of a circular  orbit around the Earth which has a period of 24 hours so it is always above the same point on the Earth’s surface . (You will need to look up the mass of the Earth.)
  2. Use the answer to (1)  to estimate what fraction of the Earth’s surface is visible at any  time from a satellite in such an orbit. (You will need to look up the radius of the Earth.)

Answers to (2) through the comments box please – and don’t forget to explain your working!