benchmarks.wiki

benchmarks.wiki / Public workspace

MMLU-Pro / 9558 / Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the Sun\operatorname{Sun} is βrE\beta r_E, where rEr_E is the radius of Earth's (assumed) circular orbit and where β<1\beta<1, show that the time the comet spends within the orbit of Earth is given by 2(1−β)⋅(1+2β)/3π×1 year \sqrt{2(1-\beta)} \cdot(1+2 \beta) / 3 \pi \times 1 \text { year } If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?
Plain-text mathematical notation (without MathML)
Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the Sun is βr_(E), where r_(E) is the radius of Earth's (assumed) circular orbit and where β<1, show that the time the comet spends within the orbit of Earth is given by
√(2(1−β))⋅(1+2β)/3π×1 year 
If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?
Original LaTeX notation
Consider a comet moving in a parabolic orbit in the plane of Earth's orbit. If the distance of closest approach of the comet to the $\operatorname{Sun}$ is $\beta r_E$, where $r_E$ is the radius of Earth's (assumed) circular orbit and where $\beta<1$, show that the time the comet spends within the orbit of Earth is given by
$$
\sqrt{2(1-\beta)} \cdot(1+2 \beta) / 3 \pi \times 1 \text { year }
$$
If the comet approaches the Sun to the distance of the perihelion of Mercury, how many days is it within Earth's orbit?

category

physics

options

  1. 82 $\text { days }$
  2. 72 days
  3. 90 $\text { days }$
  4. 110 days
  5. 45 days
  6. 85 days
  7. 64 $\text { days }$
  8. 76$ \text { days }$
  9. 100 days

src

scibench-class

Discussion

Discussion

No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

Source and history

upstream_id / Original problem ID
9558
task_id
fbada4a2-2f75-5a25-9b41-52ac5496ec9a
task_key
default--test--9558
task_revision_id
1
config / split
default / test

Official source

initial import