Average speed

[Genady]

A car goes 15 mph for 1 mile. How fast must it go in the same 1 mile back to average 30 mph for the entire round trip?

*According to a story, Albert Einstein got this puzzle in a letter from a friend and enjoyed the trick.

Edited May 21, 2023 by Genady

[swansont]

 

Spoiler

1 mile/15 mph = 1/15 of an hour (4 min)

2 miles/30 mph is also 1/15 of an hour.

The return trip must take no time, thus the speed would have to be infinite

 

[Genady]

5 minutes ago, swansont said:

 

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1 mile/15 mph = 1/15 of an hour (4 min)

2 miles/30 mph is also 1/15 of an hour.

The return trip must take no time, thus the speed would have to be infinite

 

Precisely. +1

[mistermack]

My weird mental arithmatic went like this :

Spoiler

In the dim past there was a man called "Mile a Minute Murphy" who held for a while a cycling speed record of 60mph. So it always stuck in my head that 60mph is a mile a minute. And therefore, 30 is a mile in 2 minutes, and 15 is a mile in 4 minutes. So it was obvious that the first mile takes 4 minutes, so it's impossible to do the 2 mile round trip in the 4 minutes an average of 30 would take. 

So the maths is dead simple, but I didn't immediately get the mental trick, that makes it look so obvious that going faster on the second leg will enable you to push up the overall average to past 30. For instance, do the second mile at 60, and your average goes up to 24mph, so intuitively it looks like all you need to do is go faster still to pass the 30mph average. So what's obviously happening is that as you raise the speed on the second leg, your average tends towards 30, as your speed tends towards infinity. 

 

[Genady]

2 hours ago, mistermack said:

My weird mental arithmatic went like this :

  Hide contents

In the dim past there was a man called "Mile a Minute Murphy" who held for a while a cycling speed record of 60mph. So it always stuck in my head that 60mph is a mile a minute. And therefore, 30 is a mile in 2 minutes, and 15 is a mile in 4 minutes. So it was obvious that the first mile takes 4 minutes, so it's impossible to do the 2 mile round trip in the 4 minutes an average of 30 would take. 

So the maths is dead simple, but I didn't immediately get the mental trick, that makes it look so obvious that going faster on the second leg will enable you to push up the overall average to past 30. For instance, do the second mile at 60, and your average goes up to 24mph, so intuitively it looks like all you need to do is go faster still to pass the 30mph average. So what's obviously happening is that as you raise the speed on the second leg, your average tends towards 30, as your speed tends towards infinity. 

 

Exactly. Intuition without a little arithmetic does a bad job here. +1.

(I didn't know about Murphy, and it took me longer to figure it.)

[exchemist]

9 hours ago, swansont said:

 

  Hide contents

1 mile/15 mph = 1/15 of an hour (4 min)

2 miles/30 mph is also 1/15 of an hour.

The return trip must take no time, thus the speed would have to be infinite

 

Yup, ditto. I recall these problems can be a bit counterintuitive, from trying to work the effect of the current in the Thames on a round trip in a sculling boat.  One sculls by the bank against the stream, where it is less strong, and in the centre when going with the stream. (I think I came to the conclusion that the fastest time for a round trip is in still water, since you always lose more time going against the current than you gain when going with it.)  

[Genady]

It reminded me of a puzzle with two racing bugs. One goes up and down a wall with constant speed. Another goes up, to the same height, with the half speed and then down with the double speed. Who wins?

The first bug wins. It is back to the starting point by the time the other one just gets to the top.

[Ghideon]

11 hours ago, Genady said:

A car goes 15 mph for 1 mile. How fast must it go in the same 1 mile back to average 30 mph for the entire round trip?

Trying an alternative just for fun:

Spoiler

Assume it's kind of a trick question; The problem has, from an engineering perspective, low precision meaning that there are rounding errors to account for. 30 could mean 30+-0.5? Allowing the average speed to be as low as 29.5 means the return trip speed is at least 880mph (two digit precision). Given that the current land speed record according to wikipedia is 760.343mph we can safely assume that the it is not possible for a car to drive at that speed at this time.  

 

Edited May 21, 2023 by Ghideon
Spelling

[mistermack]

33 minutes ago, Genady said:

It reminded me of a puzzle with two racing bugs.

It reminds me of a puzzle my Irish Uncle once set, when he claimed that it's impossible to break an ordinary normal glass bottle inside an ordinary empty polythene bag. It looks at first sight that it's obviously possible, but it turned out he was right. 

[Genady]

10 minutes ago, mistermack said:

It reminds me of a puzzle my Irish Uncle once set, when he claimed that it's impossible to break an ordinary normal glass bottle inside an ordinary empty polythene bag. It looks at first sight that it's obviously possible, but it turned out he was right. 

Like a garbage bag?

[mistermack]

1 minute ago, Genady said:

Like a garbage bag?

Garbage bag, plastic shopping bag etc etc.

It sounds like it should be possible, but it turns out it can't be done. 

[Genady]

10 minutes ago, mistermack said:

Garbage bag, plastic shopping bag etc etc.

It sounds like it should be possible, but it turns out it can't be done. 

Will not this work: How to Break a Glass Bottle Safely (ehow.com)?

[mistermack]

That doesn't solve the problem. It's one of those that looks ridiculously easy, but is impossible in reality.

[Genady]

16 minutes ago, mistermack said:

That doesn't solve the problem. It's one of those that looks ridiculously easy, but is impossible in reality.

 

Spoiler

Got it. If the bottle is inside the bag, the bag is not empty. 

 

Edited May 21, 2023 by Genady

[mistermack]

That's it ! You should hide your answer, and let others have a go.

[Genady]

1 minute ago, mistermack said:

That's it ! You should hide your answer, and let others have a go.

Fixed.

1 hour ago, mistermack said:

It reminds me of a puzzle my Irish Uncle once set, when he claimed that it's impossible to break an ordinary normal glass bottle inside an ordinary empty polythene bag. It looks at first sight that it's obviously possible, but it turned out he was right. 

Another bottle puzzle: 

 

[md65536]

4 hours ago, Genady said:

It reminded me of a puzzle with two racing bugs. One goes up and down a wall with constant speed. Another goes up, to the same height, with the half speed and then down with the double speed. Who wins?

The same thing happens with the relativistic Doppler effect in the twin paradox, with an outbound and return trip at the same speed. If the clocks appear to tick at 0.5x the rate of a local clock on the outbound trip, they'll appear to tick 2x on the inbound trip.

Someone once used incorrect intuition to argue on these forums that this shows that the clocks would age the same amount during the trip, thus disproving special relativity.

[Genady]

11 minutes ago, md65536 said:

The same thing happens with the relativistic Doppler effect in the twin paradox, with an outbound and return trip at the same speed. If the clocks appear to tick at 0.5x the rate of a local clock on the outbound trip, they'll appear to tick 2x on the inbound trip.

Someone once used incorrect intuition to argue on these forums that this shows that the clocks would age the same amount during the trip, thus disproving special relativity.

Hmm ... Except the twin paradox has nothing to do with the Doppler effect. It has to do with the time dilation. The latter is the same regardless of direction of the motion. It depends only on the relative speed between two frames of reference.

[TheVat]

17 hours ago, Genady said:

A car goes 15 mph for 1 mile. How fast must it go in the same 1 mile back to average 30 mph for the entire round trip?

*According to a story, Albert Einstein got this puzzle in a letter from a friend and enjoyed the trick.

Since the distance can be arbitrary, I just set it as 15 miles, and the intuitive solution popped right out at me.  

 

[Genady]

2 minutes ago, TheVat said:

Since the distance can be arbitrary, I just set it as 15 miles, and the intuitive solution popped right out at me.  

 

Oh, this is a good trick! +1

[md65536]

On 5/21/2023 at 10:08 AM, Genady said:

Hmm ... Except the twin paradox has nothing to do with the Doppler effect. It has to do with the time dilation. The latter is the same regardless of direction of the motion. It depends only on the relative speed between two frames of reference.

The relativistic Doppler effect includes time dilation, and it gives a complete solution to the basic twin paradox (hard to hide aging when you can see each other age the entire time).

https://en.wikipedia.org/wiki/Relativistic_Doppler_effect

[Genady]

4 hours ago, md65536 said:

The relativistic Doppler effect includes time dilation, and it gives a complete solution to the basic twin paradox (hard to hide aging when you can see each other age the entire time).

https://en.wikipedia.org/wiki/Relativistic_Doppler_effect

Yes, relativistic Doppler effect includes time dilation. As I know of a solution to the twin paradox that does not include relativistic Doppler effect, could you refer to a solution that does?

[md65536]

7 hours ago, Genady said:

Yes, relativistic Doppler effect includes time dilation. As I know of a solution to the twin paradox that does not include relativistic Doppler effect, could you refer to a solution that does?

Sure, apologies for veering off-topic but the ideas still relate! If you look at the Doppler factor formula, you'll see why a sign change of v gives a reciprocal factor. Before or while looking at examples, here's a challenge. Using v=+/-0.6c, the Doppler factors are 2 and 0.5 (perhaps opposite of intuition?) and Lorentz factor is 1.25. Given a statement like, "B spends 1 year traveling away while seeing A age 0.5 years. B turns around, and spends 1 year returning during which it sees A age 2 years, for total of A aging 2.5 years to B's 2," can you similarly (with no more math than that) describe what inertial twin A sees? There's no need to calculate distance, delay of light etc.

https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_doppler.html

https://hepweb.ucsd.edu/ph110b/110b_notes/node60.html

 

[Genady]

26 minutes ago, md65536 said:

Sure, apologies for veering off-topic but the ideas still relate! If you look at the Doppler factor formula, you'll see why a sign change of v gives a reciprocal factor. Before or while looking at examples, here's a challenge. Using v=+/-0.6c, the Doppler factors are 2 and 0.5 (perhaps opposite of intuition?) and Lorentz factor is 1.25. Given a statement like, "B spends 1 year traveling away while seeing A age 0.5 years. B turns around, and spends 1 year returning during which it sees A age 2 years, for total of A aging 2.5 years to B's 2," can you similarly (with no more math than that) describe what inertial twin A sees? There's no need to calculate distance, delay of light etc.

https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_doppler.html

https://hepweb.ucsd.edu/ph110b/110b_notes/node60.html

 

You are right that if we want to know what they see with their eyes, we need to use Doppler formula. And of course the result is consistent with Lorentz transform because it is the same SR.

However, to explain what happens, as opposed to what is seen, is straightforward:

In the inertial frame A, B's time is dilated 1.25 times. Thus, while B's clock has advanced by 2 years, A's clock has advanced by 2.5 years.

Edited May 23, 2023 by Genady

[joigus]

According to some quantum hype, you could do it with entanglement.  

Nice trick, @TheVat. +1

I must confess I did need a little bit of algebra.

Spoiler

If you work out the average speed in terms of the two speeds v_1, v_2, you find it's proportional to v_1*v_2/(v_1+v_2) -- the harmonic mean. That's because the overall time is prop. to 1/v_1 + 1/v_2

Some algebra shows that v_2 must be infinite

This is a cute little puzzle.