Speed of a motorcycle "around" a truck

[Yanick Toutain]

"For 10 hours the passenger of a truck sees a motorcycle circling around the truck.
The passenger calculates that the revolution speed of the motorcycle is
v = 1 km/h
But there you go... the truck is not stationary. During these ten hours, the truck drove at a speed
S = 100 km/h
The question is (obviously in relation to the road)
what is the length of the motorcycle's journey.
And so what was the average speed of the motorcycle?
And so what is the simple formula giving the value of T - S (according to S and v)

(do not ask what the radius of revolution or the period is, this data is useless to answer the question)

Subsidiary question: Does the approximate formula giving the result and/or the rigorous demonstration appear somewhere in a physics work for 3 centuries?"

[Barry Schwarz]

On Wed, 11 Oct 2023 20:59:58 -0700 (PDT), Yanick Toutain
<yanick...@gmail.com> wrote:

>"For 10 hours the passenger of a truck sees a motorcycle circling around the truck.
>The passenger calculates that the revolution speed of the motorcycle is
>v = 1 km/h
>But there you go... the truck is not stationary. During these ten hours, the truck drove at a speed
>S = 100 km/h
>The question is (obviously in relation to the road)
>what is the length of the motorcycle's journey.
>And so what was the average speed of the motorcycle?
>And so what is the simple formula giving the value of T - S (according to S and v)

The first obvious question is what is T since you never defined it.

>(do not ask what the radius of revolution or the period is, this data is useless to answer the question)
>
>Subsidiary question: Does the approximate formula giving the result and/or the rigorous demonstration appear somewhere in a physics work for 3 centuries?"

--
Remove del for email

[Yanick Toutain]

Le jeudi 12 octobre 2023 à 06:50:31 UTC+2, Barry Schwarz a écrit :
> On Wed, 11 Oct 2023 20:59:58 -0700 (PDT), Yanick Toutain
> <yanick...@.com> wrote:
>
> >"For 10 hours the passenger of a truck sees a motorcycle circling around the truck.
> >The passenger calculates that the revolution speed of the motorcycle is
> >v = 1 km/h
> >But there you go... the truck is not stationary. During these ten hours, the truck drove at a speed
> >S = 100 km/h
> >The question is (obviously in relation to the road)
> >what is the length of the motorcycle's journey.
> >And so what was the average speed of the motorcycle?
> >And so what is the simple formula giving the value of T - S (according to S and v)
> The first obvious question is what is T since you never defined it.

The main question is "And so what was the average speed of the motorcycle?"

[FromTheRafters]

After serious thinking Yanick Toutain wrote :

> "For 10 hours the passenger of a truck sees a motorcycle circling around the
> truck. The passenger calculates that the revolution speed of the motorcycle
> is v = 1 km/h
> But there you go... the truck is not stationary. During these ten hours, the
> truck drove at a speed S = 100 km/h
> The question is (obviously in relation to the road)
> what is the length of the motorcycle's journey.

Same as the truck's.

> And so what was the average speed of the motorcycle?

Along the road? the same as the truck's.

> And so what is the simple formula giving the value of T - S (according to S
> and v)

Unknown.

> (do not ask what the radius of revolution or the period is, this data is
> useless to answer the question)

Hence my 'along the road' stipulation. The MC may as well be trailered.
It is *with* the truck moving along the road.

> Subsidiary question: Does the approximate formula giving the result and/or
> the rigorous demonstration appear somewhere in a physics work for 3
> centuries?"

Unknown.

[Yanick Toutain]

Le jeudi 12 octobre 2023 à 06:50:31 UTC+2, Barry Schwarz a écrit :

Those who are destabilized by the absence of a radius in the problem statement can choose the value of their choice.
Any value will give the answer.
You can choose R = 5 / %pi (km) if you want.
By placing the motorcycle at a distance x = 1.59155 km (in front of the truck located at x = 0) at the start of the problem.

[Yanick Toutain]

[James Waldby]

Yanick Toutain ... wrote:

> "For 10 hours the passenger of a truck sees a motorcycle circling
> around the truck. The passenger calculates that the revolution
> speed of the motorcycle is v = 1 km/h
> But there you go... the truck is not stationary. During these ten
> hours, the truck drove at a speed S = 100 km/h
> The question is (obviously in relation to the road) what is the
> length of the motorcycle's journey.

If, as OP claims, radius of revolution or period isn't relevant, we
can pick any desired radius or period for doing the calculation.
Let's take r=10^6789 km as a convenient number. Suppose that at
time 0, the truck is at location (0,0) and the motorcycle is at (r,0).
Further suppose that at time 10, the truck ends up at E = (0, 1000) =
(0, 10*S). That is, we suppose the truck moved at a constant speed
along a straight line (y axis), as opposed to driving along some
curving path that could make the question more difficult to answer.

During the 10 hour trip, relative to the truck the motorcycle appears
to move at 1 km/h along a circular arc, for a total of 10 km, and a
total apparent angle alpha = 10/r = 10^(-6788) radians, ending up
at M = E + (r cos alpha, r sin alpha), which if we round off to only
a hundred decimal places is (r, 1010).

> And so what was the average speed of the motorcycle?

Rounding as before to 100 decimal places, 101 km/h.

> And so what is the simple formula giving the value of T - S (according to S and v)

A simple formula for "the value of T - S" (where T is undefined) is
"undefined".

[Yanick Toutain]

I wrote
"For 10 hours the passenger of a truck sees a motorcycle circulating around the truck.

The passenger calculates that the revolution speed of the motorcycle is
v = 1 km/h

This implies that the passenger has seen one or more revolutions
2%pi R *N = vitrevo * duration
N=1 =>1 revolution
=> R = 10*1/(2%pi) approx. 1,591 km
N=2 =>2 revolutions
=> R = 10*1/(2%pi) /2 approx 0.795 km
etc
1,591 km is the largest possible radius

Furthermore, a non-integer number of revolutions would make it impossible to state an average speed T

"And so what was the average speed of the motorcycle?

And so what is the simple formula giving the value of T - S (according to S and v)"

You tried not to give the result by transforming the problem
quote "Let's take r=10^6789 km as a convenient number."
With such a radius, the passenger would not have been able to calculate vrevo=1

[Yanick Toutain]

For 10 hours the passenger of a truck sees a motorcycle circulating around the truck.

The passenger calculates that the revolution speed of the motorcycle is
v = 1 km/h
But there you go... the truck is not stationary. During these ten hours, the truck drove at a speed
S = 100 km/h
The question is (obviously in relation to the road)
what is the length of the motorcycle's journey.

And so what was the average speed of the motorcycle T?

And so what is the simple formula giving the value of T - S (according to S and v)

(do not ask what the radius of revolution or the period is, this data is useless to answer the question

So you can choose R=5/pi km or any other value )

[James Waldby]

Yanick Toutain ... wrote:

> For 10 hours the passenger of a truck sees a motorcycle circulating
> around the truck. The passenger calculates that the revolution
> speed of the motorcycle is v = 1 km/h But there you go... the truck
> is not stationary. During these ten hours, the truck drove at a
> speed S = 100 km/h The question is (obviously in relation to the
> road) what is the length of the motorcycle's journey.

If, as OP claims, radius of revolution or period isn't relevant, we
can pick any desired radius or period for doing the calculation.
Let's take r=10^6789 km as a convenient number. Suppose that at
time 0, the truck is at location (0,0) and the motorcycle is at (r,0).
Further suppose that at time 10, the truck ends up at E = (0, 1000) =
(0, 10*S). That is, we suppose the truck moved at a constant speed
along a straight line (y axis), as opposed to driving along some
curving path that could make the question more difficult to answer.

During the 10 hour trip, relative to the truck the motorcycle appears
to move at 1 km/h along a circular arc, for a total of 10 km, and a
total apparent angle alpha = 10/r = 10^(-6788) radians, ending up at M

= E + (r cos alpha, r sin alpha), which if we round off to a few

hundred decimal places is (r, 1010).

> And so what was the average speed of the motorcycle T?

1010/10, or 101 km/h

> And so what is the simple formula giving the value of T - S (according to S and v)

In terms of truck speed S, motorcycle average speed T, rotation
velocity v, and elapsed time w, we have T = (w*S + w*v)/w = S + v, so
that T-S = v = 1 km/h

> (do not ask what the radius of revolution or the period is, this
> data is useless to answer the question So you can choose R=5/pi km
> or any other value )

As noted above, I chose 10^6789 km as a convenient number.

> Subsidiary question: Does the approximate formula giving the result
> and/or the rigorous demonstration appear somewhere in a physics work
> for 3 centuries?

This "subsidiary question" isn't clear. By "appear somewhere in a
physics work for 3 centuries" do you mean something appearing for any
term of exactly 300 years, no more, no less, or do you mean coinciding
exactly with calendar centuries, eg from the beginning of 1 Jan 500 to
the end of 31 Dec 799? Does the 300 year term you are thinking of
account for the fact that years that are multiples of 400 are leap
years, but that other multiple-of-100 years are not? Are you going to
deduct for leap-seconds? And do you have some mechanism in mind that
will make something disappear after it exists for exactly 300 years?
You should clarify your question, eg specify how many seconds you
actually mean, what the conditions are on the starting second, what is
meant by "appear", by "physics", by "work", etc.

[FromTheRafters]

Yanick Toutain pretended :

If the radius is zero, it *is* the truck, that is, the origin of your
circle, no?

[Yanick Toutain]

Quote "the passenger of a truck sees a motorcycle circling around the truck "
The radius is not zero

[Yanick Toutain]

AGAIN NO

Quote "1,591 km is the largest possible radius"

The passenger calculates that the revolution speed of the motorcycle is
v = 1 km/h

This implies that the passenger has seen one or more revolutions
2%pi R *N = vitrevo * duration
N=1 =>1 revolution
=> R = 10*1/(2%pi) approx. 1,591 km
N=2 =>2 revolutions
=> R = 10*1/(2%pi) /2 approx 0.795 km
etc
1,591 km is the largest possible radius

[James Waldby]

Are you claiming you previously stated some "largest possible radius"?
You didn't say anything about such a limit.

> The passenger calculates that the revolution speed of the motorcycle
> is v = 1 km/h This implies that the passenger has seen one or more
> revolutions

It implies no such thing. The passenger merely needs to measure
whatever angle the motorcycle moves in some measured interval at a
measured radius, and do the proper arithmetic to get relative speed.
The stated problem does not say or imply anything about completed
revolutions.

> 2%pi R *N = vitrevo * duration N=1 =>1 revolution => R = 10*1/(2%pi)
> approx. 1,591 km
> N=2 =>2 revolutions => R = 10*1/(2%pi) /2 approx 0.795 km etc
> 1,591 km is the largest possible radius

These examples are misleading due to inconsistent usage of comma vs
period as decimal point or thousands indicator.

>> > Subsidiary question: Does the approximate formula giving the result
>> > and/or the rigorous demonstration appear somewhere in a physics work
>> > for 3 centuries?

Don't forget to clarify that question, as previously requested:

[Yanick Toutain]

Le jeudi 19 octobre 2023 à 04:50:09 UTC+2, James Waldby a écrit :

Your behavior resembles that of a student who slept in class throughout the first term and who, unable to do an exercise (and refusing to revise to catch up on his shortcomings) disrupts the class by inventing a thousand and one reasons which - according to him - would make it impossible to resolve the problem.
When I write to you that the passenger has seen at least one revolution in 10 hours, this definitively caps the radius at 5/%pi
(to announce a revolution speed, at least one revolution is required)

[FromTheRafters]

Yanick Toutain presented the following explanation :

Quote "Any value will give the answer"

[Yanick Toutain]

Radius having any value... compatible with the rest of the problem.
it is obvious

no researcher has ever imagined the biker inside the cab of the truck turning around the passenger with his motorcycle or even the biker being so far away that he could not make a single complete revolution... to prevent the passenger from knowing its ... speed of revolution

[Mike Terry]

You don't need the biker to have made a complete revolution of the truck in order to measure the
biker's speed of revolution around the truck. Just like a car does not need to travel for an hour
in order to measure its speed in miles per hour.

Mike.