dull problem

[John Hanna]

Spotted a question on fb “Dull Men’s Club”:

This maybe too dull or not dull enough but the thought experiment really intrigues me. Here goes.

What would be the theoretical fastest time that passengers could get from Sydney to London, a distance of about 17,000Km (10,563 miles), shortest direct route. Assuming an aircraft/rocket could fly at unlimited speed. So on the journey the vehicle would keep accelerating to the halfway point and then decelerate for the other half.

The passengers would have to put up with the constant acceleration g force which must not be too uncomfortable for them. I’m guessing, probably wrongly, that about 2g may be the maximum.

What would the be the time taken and what would be the maximum speed reached. I know it’s a mathematical problem and one I don’t have the skills to solve. Hopefully someone here may be clever enough to know how to work it out.

 

(there’s also a link to a Reddit post provided but I don’t have an account.)

 

Did some mathematics on TI-Nspire (attached). Started with a(t)=2g:

I divided 17000000m by 2 assuming accelerating halfway and then decelerating to get 931.315s for half the trip.

 

 

I then asked Microsoft’s Copilot (AI) the same question:

It replied (copied from Copilot and I dislike the formatting!):

Let’s calculate the time it would take to travel 17,000 km with a constant acceleration of 2g (where (g) is the acceleration due to gravity).

1. First, let’s find the acceleration in meters per second squared:
• (a = 2 \times g = 2 \times 9.80665 , \text{m/s}^2 \approx 19.61 , \text{m/s}^2).
2. Next, we can use the kinematic equation to find the time ((t)):

[ d = \frac{1}{2} \cdot a \cdot t^2 ]

Solving for (t):

[ t = \sqrt{\frac{2 \cdot d}{a}} ]

Plugging in the values: [ t = \sqrt{\frac{2 \cdot 17,000,000 , \text{m}}{19.61 , \text{m/s}^2}} ]

Calculating: [ t \approx 931 , \text{seconds} ]

3. Converting seconds to minutes:
• (931 , \text{seconds} \approx 15.5 , \text{minutes}).

Therefore, it would take approximately 15.5 minutes to travel 17,000 km with a constant acceleration of 2g.

 

But Copilot calculates the entire trip would take 931s. Why the discrepancy???

What formatting ‘language’ is Copilot using??? It’s not html… LaTex?

 

Regards,

     John

 

[John Hanna]

See attachment this time 😊

 

Regards,

     John

 

[Karlheinz Haas]

John,

If I assume the acceleration to be 19.6 m/s^2 for the entire distance (as I thought Copilot may have done) I get a time of t=1,317.1 sec 

For half the distance, I get your value for t and a max. velocity after that time of 18,440 m/s, with an average velocity of 9,220 m/s. 

To check our math, I plugged that back into d=v(avg)* t and got the original total distance. 

So, looks like you did it correctly, just not sure how Copilot is "reading" the prompt.

There's a great website that lets you input the cities and draws the great circle arc most likely to be (close to) the flight route. And as you can see, this one actually gave us the 17,000 km distance.

Karlheinz

[Karlheinz Haas]

Not sure why the screenshots didn't work. Wrong format, I guess.

Let's try again, John.

If I assume the acceleration to be 19.6 m/s^2 for the entire distance (as I thought Copilot may have done) I get a time of t=1,317.1 sec 

For half the distance, I get your value for t and a max. velocity after that time of 18,440 m/s, with an average velocity of 9,220 m/s. 

To check our math, I plugged that back into d=v(avg)* t and got the original total distance. 

So, looks like you did it correctly, just not sure how Copilot "read" the prompt.

There's a great website that lets you input the cities and draws the great circle arc most likely to be (close to) the flight route. And as you can see, this one actually gave us the 17,000 km distance.

Karlheinz