Significant Figures Verification

In summary: That's a lot more work than is worth doing for this problem.In summary, the provided solution for the problem of an asteroid falling to Earth from a great distance uses an incorrect number of significant figures for the kinetic energy imparted to the planet. The correct answer should have only one significant figure, but the solution reports three significant figures. This may be due to a questionable assumption of three significant figures in the mass of the asteroid. However, in cases where the first digit of the result is a 1 and the least significant inputs have first digits of 8 or 9, it may be appropriate to add an extra significant digit to the result.
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TRB8985
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Homework Statement
The gravitational pull of the Earth on an object is inversely proportional to the square of the distance of the object from the center of the Earth. At the Earth’s surface, this force is equal to the object’s normal weight, mg, where 𝑔 = 9.8 𝑚/𝑠^2 , and at large distances, the force is zero.

If a 20,000 kg asteroid falls to Earth from a very great distance away, what will be its minimum speed as it strikes the Earth’s surface, and how much kinetic energy will it impart to our planet? You can ignore the effects of the Earth’s atmosphere.
Relevant Equations
Kinetic energy imparted to planet = m_asteroid * g * radius_Earth
Good morning,

I've completed the problem provided above and have verified my answers are correct, but I'm running into a strange situation when it comes to the solution's answer for the kinetic energy portion.

For the kinetic energy imparted to the planet, we're taking 20,000 kg * 9.8 m/s² * 6.371E6 m. This completely matches the solution's answer.

I'm aware that there's one significant figure in 20,000, two in 9.8, and 4 in 6.371E6. My understanding is that when multiplying these values together, the result should be reported using the lowest number of significant figures in these three values - so just one, 1E12 J.
The official solution for this problem reports three, however - 1.25E12 J.

Am I making a mistake in my reporting of the answer? Or is the provided solution using the incorrect number of significant figures here?

Thank you!
 
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  • #2
TRB8985 said:
Homework Statement:: The gravitational pull of the Earth on an object is inversely proportional to the square of the distance of the object from the center of the Earth. At the Earth’s surface, this force is equal to the object’s normal weight, mg, where 𝑔 = 9.8 𝑚/𝑠^2 , and at large distances, the force is zero.

If a 20,000 kg asteroid falls to Earth from a very great distance away, what will be its minimum speed as it strikes the Earth’s surface, and how much kinetic energy will it impart to our planet? You can ignore the effects of the Earth’s atmosphere.
Relevant Equations:: Kinetic energy imparted to planet = m_asteroid * g * radius_Earth

Good morning,

I've completed the problem provided above and have verified my answers are correct, but I'm running into a strange situation when it comes to the solution's answer for the kinetic energy portion.

For the kinetic energy imparted to the planet, we're taking 20,000 kg * 9.8 m/s² * 6.371E6 m. This completely matches the solution's answer.
The energy for a fall from infinity to the surface under inverse square matches the energy for a fall from surface to center at constant surface gravity. Yes, that would be a correct result.

TRB8985 said:
I'm aware that there's one significant figure in 20,000, two in 9.8, and 4 in 6.371E6. My understanding is that when multiplying these values together, the result should be reported using the lowest number of significant figures in these three values - so just one, 1E12 J.
I agree. One significant figure.
TRB8985 said:
The official solution for this problem reports three, however - 1.25E12 J.
Unless the asteroid's mass has additional significant figures, I agree that this is too many significant figures.
TRB8985 said:
Am I making a mistake in my reporting of the answer? Or is the provided solution using the incorrect number of significant figures here?
I agree with you and disagree with the book.

However, if one ascribes (questionably) three significant figures to that 20,000 kg then an argument can be made that three digits is appropriate in the result:

The maximum relative error in 9.8 (two sig figs) is only about one part in two hundred. Less than one part in one thousand in reality.

The relative error in 1.2 (two sig figs) is almost one part in twenty. About one part in 24 in this case.

Do we really want to be over-reporting the unreliability of the result by a factor of 4 (maximum error) or by a factor of 40 (actual error).

When presenting a result whose first digit is a 1 and whose least significant inputs have first digits of 8 or 9, I would tend to err on the side of adding one more significant digit in the result than the rules call for.

When adding large columns of numbers, I would tend to err on the side of subtracting one significant digit from the result. One more for every factor of 100 in the number of entries being totalled to account for statistical noise in the rounding error. But that's enough work that it may be worth doing a real error analysis at that point.
 
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Related to Significant Figures Verification

What are significant figures and why are they important?

Significant figures are the digits in a number that carry meaning or contribute to the precision of the measurement. They are important because they indicate the level of accuracy of a measurement and help maintain consistency in calculations.

How do you determine the number of significant figures in a given number?

The general rule for determining significant figures is to count all non-zero digits and any zeros between non-zero digits. For example, the number 150 has three significant figures, while 0.005 has one significant figure. However, there are specific rules for certain cases, such as trailing zeros after a decimal point or leading zeros before a non-zero digit.

Why is it necessary to verify significant figures in calculations?

When performing calculations, the result should have the same number of significant figures as the least precise measurement used in the calculation. Verifying significant figures ensures that the final result is accurate and not falsely precise.

What is the process for rounding a number to the correct number of significant figures?

To round a number to the correct number of significant figures, follow these steps: 1. Identify the digit in the last significant figure. 2. If the next digit is 5 or greater, round up the last significant digit. 3. If the next digit is less than 5, leave the last significant digit as is. 4. If the next digit is exactly 5, round up if the last significant digit is odd, or leave as is if the last significant digit is even.

How do significant figures affect scientific measurements and data analysis?

Significant figures play a crucial role in scientific measurements and data analysis. They help determine the precision and accuracy of measurements, and ensure that results are not falsely precise. In data analysis, significant figures are used to indicate the level of uncertainty in the data and are important in making accurate conclusions and predictions.

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