Practical Skills - Recording time with human error

[Jimmy87]

Homework Statement

Please could someone help me with how raw data should be dealt with when using a human operated stop watch. I have attached the information we were given and it makes little sense to me.

Homework Equations

See attachement

The Attempt at a Solution

We are constantly reminded that all columns in a results table which are calculated should be to the same number of significant figures but all measured values should be to the same number of decimal places, in line with the equipment used. The problem is when using a stop watch. We are told NOT to use the raw values from the stop watch because they measure to the nearest 0.01s and human reaction times are about 0.1s. In the attached document it says to round down to the nearest tenth of a second which makes sense as the reaction time is always going to be a delayed reaction time. If you do what they say in the document you do not get the values for 'T' in the second column. For example, the first value is 10.49s and if you round down to 10.4 and divide by 20, you get exactly 0.52. Is this just a mistake or am I missing something? Also, it really doesn't make sense that you end up with a value of T which is more precise than the equipment measures itself!? Is that really right? The first value for T is 0.525 which the stop watch could not measure if you were to measure T rather than calculate it?

Thanks for any help!

 

[DrClaude]

For example, the first value is 10.49s and if you round down to 10.4 and divide by 20, you get exactly 0.52. Is this just a mistake or am I missing something?

It is better to keep non-significant digits in all intermediate calculations, and only round off at the end.

Also, it really doesn't make sense that you end up with a value of T which is more precise than the equipment measures itself!? Is that really right? The first value for T is 0.525 which the stop watch could not measure if you were to measure T rather than calculate it?

Lets take an extreme example: imagine that there were 1 million oscillations in that 10.49 seconds. Would you still think that you can't know the period of a single oscillation to better than 0.1 s?

 

[Jimmy87]

It is better to keep non-significant digits in all intermediate calculations, and only round off at the end.Lets take an extreme example: imagine that there were 1 million oscillations in that 10.49 seconds. Would you still think that you can't know the period of a single oscillation to better than 0.1 s?

Thanks DrClaude that has helped a lot. How would you keep significant figures constant in a calculated column if your answer is less than the number you are wanting to use. What we are told is that calculated values should be reported to the same number of significant figures as the lowest number used in the calculation. What happens if the lowest number used is 3 but the calculator gives an answer with 2. For example, if T in that question was actually 10.4 then divided by 20 would give 0.52. Wuld you just add an extra zero? So 0.520. If your dividing by a fixed number (20 in this case) does that count as the quantity with the lowest number of s.f.? So should you actually report to two s.f.? An extreme example could be 10.49 / 10.49 = 1 so should this be 1.000 since both quantities in the calculation are to 4 s.f.?

Thanks again

 

[DrClaude]

What happens if the lowest number used is 3 but the calculator gives an answer with 2. For example, if T in that question was actually 10.4 then divided by 20 would give 0.52. Wuld you just add an extra zero? So 0.520.

Yes. Note that while this is unambiguous to the right of the decimal point, it can be ambiguous to the left. For instance, how do you report 5201 to 3 significant figures? The best way is to use scientific notation: 5.20 × 10³.

If your dividing by a fixed number (20 in this case) does that count as the quantity with the lowest number of s.f.? So should you actually report to two s.f.?

Integers are assumed to have infinite precision. If you counted 20 oscillations, it is exactly 20, not 19, not 21, not 20.1.

An extreme example could be 10.49 / 10.49 = 1 so should this be 1.000 since both quantities in the calculation are to 4 s.f.?

Here the 1 is not an integer but a real number, which is simply close within measuring error to an integer so 1.000.

Just to be clear, what I mean by integer here refer to things that can be counted.

 

[Jimmy87]

Yes. Note that while this is unambiguous to the right of the decimal point, it can be ambiguous to the left. For instance, how do you report 5201 to 3 significant figures? The best way is to use scientific notation: 5.20 × 10³.Integers are assumed to have infinite precision. If you counted 20 oscillations, it is exactly 20, not 19, not 21, not 20.1.Here the 1 is not an integer but a real number, which is simply close within measuring error to an integer so 1.000.

Just to be clear, what I mean by integer here refer to things that can be counted.

Thanks for your help DrClaude. Just one last question I was thinking of that is bugging me. I get that if 20T is measured to 10.49 then T will end up to 3 decimal places since we physically counted the oscillations. However, what about this example. Say you measure the time it takes a car to move a certain distance. It moves 10.49m in 10.49s. Both of these are now measured values and there is not a counted value. If you work out the speed you get 1 m/s. According to all the rules I should report the answer to the same number of s.f. as the lowest quantity. They both have 4 s.f. so 1.000 m/s. This means in 1s it goes 1.000m. We now know the distance to more precision than the instrument that was used to measure the distance originally?

 

[DrClaude]

They both have 4 s.f. so 1.000 m/s. This means in 1s it goes 1.000m. We now know the distance to more precision than the instrument that was used to measure the distance originally?

You are assuming here that the speed of car was constant, which is a very big assumption! But even granting this, what you are deducing is the position of the car with respect to where it was at time t = 0 s. With what precision do you know that point?