The Work Done Against Gravity 3. The magnitude of the force of Earth's gravity on an object of mass m is given by F(x) = m a(x) = GmME R 1 (1+x/RE)² When a force F(x) is applied to an object, the work done by the force is defined as W = [² FC F(x) dx where the object begins at position x = a and ends at position x = b. (a) Write the force of gravity F(x) as a power series. (Hint: Use the power series for a(x) from problem 2.) (b) Determine the work done against the force of gravity from x = 0 to x = h as a series by integrating the power series F(x). h is the final height of the object. (c) The work done against gravity by moving the object from x = 0 to x = h is called the potential energy, U. Determine the first order Taylor polynomial, U≈ T₁(h), GME for the potential energy. Simplify the constant just as you did for part (f) R²/ of problem 2. This formula for U should look familiar if you have taken Physics I.

The Work Done Against Gravity 3. The magnitude of the force of Earth's gravity on an object of mass m is given by F(x) = m a(x) = GmME R 1 (1+x/RE)² When a force F(x) is applied to an object, the work done by the force is defined as W = [² FC F(x) dx where the object begins at position x = a and ends at position x = b. (a) Write the force of gravity F(x) as a power series. (Hint: Use the power series for a(x) from problem 2.) (b) Determine the work done against the force of gravity from x = 0 to x = h as a series by integrating the power series F(x). h is the final height of the object. (c) The work done against gravity by moving the object from x = 0 to x = h is called the potential energy, U. Determine the first order Taylor polynomial, U≈ T₁(h), GME for the potential energy. Simplify the constant just as you did for part (f) R²/ of problem 2. This formula for U should look familiar if you have taken Physics I.

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter3: Polynomial Functions

Section3.5: Mathematical Modeling And Variation

Problem 7ECP: The kinetic energy E of an object varies jointly with the object’s mass m and the square of the...

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Question

See the 2nd image when it says to referance the power series from question 2.

The Work Done Against Gravity
3. The magnitude of the force of Earth's gravity on an object of mass m is given by
F(x) = m a(x) =
GmME
R
1
(1+x/RE)²
When a force F(x) is applied to an object, the work done by the force is defined as
W
= [² FC
F(x) dx
where the object begins at position x = a and ends at position x = b.
(a) Write the force of gravity F(x) as a power series. (Hint: Use the power series for
a(x) from problem 2.)
(b) Determine the work done against the force of gravity from x = 0 to x = h as a
series by integrating the power series F(x). h is the final height of the object.
(c) The work done against gravity by moving the object from x = 0 to x = h is called
the potential energy, U. Determine the first order Taylor polynomial, U≈ T₁(h),
GME
for the potential energy. Simplify the constant just as you did for part (f)
R²/
of problem 2. This formula for U should look familiar if you have taken Physics
I.

Newton's Law of Gravitation
2. The magnitude of the acceleration of an object under the pull of Earth's gravity is
given by Newton's Universal Law of Gravitation
a = G
ME
R²
where G is the universal gravitational constant, Me is the mass of Earth, and R is the
distance of the object from the center of Earth.
Let x be the distance above Earth's surface. We can rewrite the formula for the
acceleration as a function of x by noting that R = RẺ + x, where RE is the radius of
Earth. Therefore,
a(x) = G₂
ME
(RE + x)²

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