Calculating Speed from Intervals: Feynman's Lecture on Physics Vol. 1

[rudransh verma]

"We know only where she was at intervals of one minute from the table; we can get a rough idea that she was going 5000 ft/min during the 7th minute, but we do not know, at exactly the moment 7 minutes, whether she had been speeding up and the speed was 4900 ft/min at the beginning of the 6th minute, and is now 5100 ft/min, or something else, because we do not have the exact details in between. So only if the table were completed with an infinite number of entries could we really calculate the velocity from such a table."
From feynmans lectures on physics. Vol 1.

Isnt it should be during the 6th minute?
And in the bold , does it talk about the average speed which is firstly 4900 during 6th minute and then 5100 at the end of 6th minute.

 

[Steve4Physics]

"We know only where she was at intervals of one minute from the table; we can get a rough idea that she was going 5000 ft/min during the 7th minute

The 1st minute is the time-interval from t=0 to t=1min.
The 7th minute is the time-interval from t=6min to t=7min.
From the table, the average speed durig the 7th minutes is 5000ft/s. So the reference to the “7th minute” is correct.

, but we do not know, at exactly the moment 7 minutes, whether she had been speeding up and the speed was 4900 ft/min at the beginning of the 6th minute, and is now 5100 ft/min, or something else, because we do not have the exact details in between. So only if the table were completed with an infinite number of entries could we really calculate the velocity from such a table."
From feynmans lectures on physics. Vol 1.

I also find that a little confusing. Maybe “at the beginning of the 6th minute” should actually say “at the beginning of the 7th minute”. That would make more sense.

 

[jbriggs444]

we do not know, at exactly the moment 7 minutes, whether she had been speeding up and the speed was 4900 ft/min at the beginning of the 6th minute, and is now 5100 ft/min, or something else

I also find that a little confusing. Maybe “at the beginning of the 6th minute” should actually say “at the beginning of the 7th minute”. That would make more sense.

Confusing how? We are at t=7 minutes. This is at the end of the 7th minute, not the beginning. The average speed during the preceding minute (the seventh minute) was 5000 feet per minute.

This could have been achieved with uniform acceleration from v=4900 at t=6 to v=5100 with t=7.

If we were to assume that position is continuously differentiable (i.e. that position is continuous, that it has a first derivative that is defined everywhere and called "velocity" then we can apply the mean value theorem:

Let $\displaystyle f:[a,b]\to \mathbb {R}$ be a continuous function on the closed interval $\displaystyle [a,b]$, and differentiable on the open interval $\displaystyle (a,b)$, where $\displaystyle a<b$. Then there exists some $\displaystyle c$ in $\displaystyle (a,b)$ such that

$$\displaystyle f'(c)={\frac {f(b)-f(a)}{b-a}}$$

Correlary: If the average velocity is 5000 feet per minute for the 7th minute then the instantaneous velocity must be exactly 5000 feet per minute at some time during the 7th minute (i.e. between t=6 minutes and t=7 minutes).

 

[rudransh verma]

Confusing how? We are at t=7 minutes. This is at the end of the 7th minute, not the beginning. The average speed during the preceding minute (the seventh minute) was 5000 feet per minute.

I don’t get this paragraph!

This could have been achieved with uniform acceleration from v=4900 at t=6 to v=5100 with t=7.

That is what @Steve4Physics is saying ie during/beginning of the 7th minute.

 

[Steve4Physics]

Confusing how? We are at t=7 minutes. This is at the end of the 7th minute, not the beginning. The average speed during the preceding minute (the seventh minute) was 5000 feet per minute.

The original passage (Post #1) specifically refers to “the beginning of the 6th minute”, i.e. t=5min:
"the speed was 4900 ft/min at the beginning of the 6th minute"

A clearer explanation (in my opinion) would have been to use only:
- the instantaneous speed at t=6min;
- the instantaneous speed at t=7min;
- the average speed during this interval (i.e. the7th minute).
The speed could have been 4900 ft/min at t=6min and 5100 ft/min at t=7min.

The confusion (maybe 'lack of clarity' is better) is the use of the instantaneous speed at t=5min - which is outside the period of interest (the 7th minute). For a clear explanation, we do not want to have to consider what happens in the 6th and 7th minutes when we only need to consider what happens in the 7th minute.

Just my opinion though!

 

[jbriggs444]

I don’t get this paragraph!

That is what @Steve4Physics is saying ie during/beginning of the 7th minute.

- the instantaneous speed at t=6min;

Ahh, gotcha. I read right past "beginning of 6th minute", interpreting it as t=6. As the writer likely intended but did not correctly convey.

 

[rudransh verma]

Ahh, gotcha. I read right past "beginning of 6th minute", interpreting it as t=6. As the writer likely intended but did not correctly convey.

So it is incorrect to say in the beginning of 6th minute. Should be 7th minute?

Also If there is a car moving we find its speed suppose by measuring how far it traveled in 1sec. That is a correct way to calculate speed instead of talking how far it traveled in an hour or a year if the car has suppose traveled only for 7 minutes. So this speed is 160km/sec when measured from t=5sec to t=6sec. How can we know if its the speed at 5th sec or 6th sec.(Avoid derivatives)

 

[jbriggs444]

So it is incorrect to say in the beginning of 6th minute. Should be 7th minute?

Also If there is a car moving we find its speed suppose by measuring how far it traveled in 1sec. That is a correct way to calculate speed instead of talking how far it traveled in an hour or a year if the car has suppose traveled only for 7 minutes. So this speed is 160km/sec when measured from t=5sec to t=6sec. How can we know if its the speed at 5th sec or 6th sec.(Avoid derivatives)

The units were feet and minutes. How did we get to kilometers and seconds? And how did we get to 160 distance units traversed during the one minute between t=5 to t=6? The chart says 340 feet.

As I understand it, you want to know from the stated position measurements at t=0, t=1, t=2, etc the instantaneous speed at t=5 or at t=6. You cannot determine either answer from the given information.

 

[rudransh verma]

As I understand it, you want to know from the stated position measurements at t=0, t=1, t=2, etc the instantaneous speed at t=5 or at t=6. You cannot determine either answer from the given information.

Why? Take the interval 6th minute and take the ratio of the distance upon time and then take limit. We will get the speed at 5sec. Isn’t it?

 

[Steve4Physics]

Why? Take the interval 6th minute and take the ratio of the distance upon time and then take limit. We will get the speed at 5sec. Isn’t it?

No. Also, you are mixing minutes and seconds which adds to the confusion.

You can't determine an exact instantaneous speed from a table of times and distances. If you think you can, give a worked example to illustrate your method.

 

[rudransh verma]

No. Also, you are mixing minutes and seconds which adds to the confusion.

You can't determine an exact instantaneous speed from a table of times and distances. If you think you can, give a worked example to illustrate your method.

In post #7 that is a different question actually. what if we are given an infinite number of entries in the table?

 

[jbriggs444]

In post #7 that is a different question actually. what if we are given an infinite number of entries in the table?

If there are an infinite number of entries and if they are dense in the domain of the function and if the function is continuous then those entries completely determine the function. The instantaneous velocity is defined as the derivative of the function.

So those infinitely many table entries would determine the instantaneous velocity at 5 minutes exactly.

However, if you want a finite procedure to take those entries and calculate the derivative then you are out of luck. There is no finite procedure to calculate a derivative from a table of function values.

 

[rudransh verma]

There is no finite procedure to calculate a derivative from a table of function values.

You mean by the derivative “instantaneous rate or the limit of the ratio ”
That we cannot find rate or derivatives by just looking at the table. We need to determine the function first?
What if we are given values of position at times like 0.001, 0.0000001, 0.000000000000000001 sec. Then we can find ratio of distance/time and we can see where that converges. That will be the instantaneous rate I guess?

 

[jbriggs444]

You mean by the derivative “instantaneous rate or the limit of the ratio ”

Yes. Both phrasings work and mean the same thing.

That we cannot find rate or derivatives by just looking at the table. We need to determine the function first?
What if we are given values of position at times like 0.001, 0.0000001, 0.000000000000000001 sec. Then we can find ratio of distance/time and we can see where that converges. That will be the instantaneous rate I guess?

Yes. We cannot calculate derivatives from tables. What we can calculate are average velocities. In principle, if you were to calculate average velocities over times like 0.001, 0.0000001, 0.000000000000000001 and so on, the averages you compute would get as close to the derivative as you please.

Indeed, this is how the derivative is defined. It is the limit that is approached (if one is approached) by the average rate as the time interval is reduced toward zero.

Note that experimental error becomes a bigger and bigger problem as one tries to focus in on smaller and smaller time intervals. In principle we can calculate better and better approximations to the derivative. In practice, we cannot.