Problem 2: We discussed gravitational acceleration near the ground to be a constant g, which gives the position of an object in free fall as quadratic in time. Let's say that, in the remote future, a cataclysmic event has destabilised the Earth's interior, and the planet is disintegrating. The gravity near the surface of the collapsing Earth is then found as: Iomt (t) = ge¬at² where t > 0 is time, a 2 0 is some constant, and g is the original surface gravitational acceleration. During the cataclysm, a small rock was ejected straight off the surface at some speed vo at time t = 0. The near-surface gravity, however, was still strong enough that it started pulling the rock back. Unlike the constant surface gravity case, however, notice lim-∞ Gomt (t) = 0, implying that this rock will have a terminal velocity it will settle into. What is the rock's terminal speed v»? When calculating væ, you may run into a definite integral that is not easily solvable. Fortunately, this is a well-known and important result in STEM. You may search up this integral, and write a brief paraphrase of how your source solves the integral. BTW, the indefinite counterpart of the integral is not known as a closed form, so make sure you have a definite integral set up.
Problem 2: We discussed gravitational acceleration near the ground to be a constant g, which gives the position of an object in free fall as quadratic in time. Let's say that, in the remote future, a cataclysmic event has destabilised the Earth's interior, and the planet is disintegrating. The gravity near the surface of the collapsing Earth is then found as: Iomt (t) = ge¬at² where t > 0 is time, a 2 0 is some constant, and g is the original surface gravitational acceleration. During the cataclysm, a small rock was ejected straight off the surface at some speed vo at time t = 0. The near-surface gravity, however, was still strong enough that it started pulling the rock back. Unlike the constant surface gravity case, however, notice lim-∞ Gomt (t) = 0, implying that this rock will have a terminal velocity it will settle into. What is the rock's terminal speed v»? When calculating væ, you may run into a definite integral that is not easily solvable. Fortunately, this is a well-known and important result in STEM. You may search up this integral, and write a brief paraphrase of how your source solves the integral. BTW, the indefinite counterpart of the integral is not known as a closed form, so make sure you have a definite integral set up.
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter13: Gravitation
Section: Chapter Questions
Problem 13.3CYU: Check Your Understanding Why not use the simpler expression U=mg(y2y1) ? How significant would the...
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Need help with prob 2. Please show your work and box your answer
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Transcribed Image Text:Problem 2: We discussed gravitational acceleration near the ground to be a constant g, which gives the
position of an object in free fall as quadratic in time. Let's say that, in the remote future, a cataclysmic
event has destabilised the Earth's interior, and the planet is disintegrating. The gravity near the surface of
the collapsing Earth is then found as:
Iomt(t) = ge¬at²,
where t > 0 is time, a > 0 is some constant, and g is the original surface gravitational acceleration.
During the cataclysm, a small rock was ejected straight off the surface at some speed vo at time t = 0. The
near-surface gravity, however, was still strong enough that it started pulling the rock back.
Unlike the constant surface gravity case, however, notice lim→∞0 Jomt (t) = 0, implying that this rock will
have a terminal velocity it will settle into. What is the rock's terminal speed v»?
When calculating v0, you may run into a definite integral that is not easily solvable. Fortunately, this is a
well-known and important result in STEM. You may search up this integral, and write a brief paraphrase
of how your source solves the integral. BTW, the indefinite counterpart of the integral is not known as a
closed form, so make sure you have a definite integral set up.
BONUS: Numerically (i.e. using software) compute and make a plot of v(t).
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