(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant acceleration of 0.170 m/s^{2}.

For how much time and what distance does the student have to run at 5.0 m/s before she overtakes the bus?

2. Relevant equations

x = ∫vdt

v = ∫adt

Quadratic formula

3. The attempt at a solution

x_{1}=5.0t

x_{2}=40.0 + 2*0.170*t^{2}

The person and bus meet when x_{1}=x_{2}, so to solve for t, I rewrite as a quadratic equal to zero and use the formula:

5.0t = 40.0 + 0.0850*t^{2}→ 0.0850*t^{2}- 5.0t + 40.0 = 0

t = [itex]\frac{5.0 \pm \sqrt{(5.0)^{2} - 4(0.0850)(40.0)}}{2(0.0850)}[/itex]

So here is where my question actually comes in. How do I solve this using the appropriate number of significant digits? My book says the answers are 9.55 and 49.3, which you get if you calculate the exact answer, and then use 3 significant digits, but why 3 in this case?

My book gives significant digit rules for addition/subtraction and multiplication/division, but does not say anything about equations using both types of operations. I have thoroughly searched online resources and actually find that most sites out there use slightly different methods, and several I referenced actually contradicted themselves or miscalculated their own examples based on their own rules. It doesn't seem like this should be that confusing.

If I follow my interpretation of my book's rules, here's what I get:

1. Because 5.0 has two significant digits, 5.0*5.0=25

2. Because 4 is an exact number in the formula it is ignored when determining significant digits, and 4*0.0850*40.0 = 13.6, since both 0.0850 and 40.0 have three significant digits.

3. Because 2 is an exact number in the formula it is ignored when determining significant digits, and 2*0.0850 = 0.170, since 0.0850 has three significant digits.

t = [itex]\frac{5.0 \pm \sqrt{25-13.6}}{0.170}[/itex]

4. Because 25 is the most uncertain numbers, 25 - 13.6 = 11

t = [itex]\frac{5.0 \pm \sqrt{11}}{0.170}[/itex]

5. Because 11 has two significant digits, sqrt(11) = 3.3

t = [itex]\frac{5.0 \pm 3.3}{0.170}[/itex]

6. Because both 5.0 and 3.3 have the same uncertainty, the numerator becomes 8.3 and 1.7.

7. Because both 8.3 and 1.7 have two significant digits, the two of them divided by the denominator would result in two significant digit answers:

t = 49 and t = 10

Now, I know that I'm compounding the rounding, which is leading my answer away from more accurate values, but isn't that the point?

MANY, many, many sites have told me to perform the equation as a whole first, and then round the final answer, but none of them explain how to determine the number of significant digits to use when doing it that way... I'm a bit disappointed that my book doesn't address how to handle significant figures for equations containing both adding/subtraction and multiplication/division operations. It would appear that if all the variables had the same number of significant digits, then the answer would too, but in this case, one of the variables has 2, and the others have 3.

Thanks for any help you all can provide! It is greatly appreciated!

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# How to determine significant figures for quadratic?

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