Ch. 06 - Gravitation and Newton's Synthesis

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

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Giancoli Douglas 5th edition

Ch. 06 - Gravitation and Newton's Synthesis

Problem 79

Chapter 6, Problem 79

A particle is released at a height r_E (radius of Earth) above the Earth’s surface. Determine its velocity when it hits the Earth. Ignore air resistance. [Hint: Use Newton’s second law, the law of universal gravitation, the chain rule, and integrate.]

Verified step by step guidance

Start by identifying the forces acting on the particle. The only force acting is the gravitational force, given by Newton's law of universal gravitation:

F = \frac{G m M}{r^{2}}

, where

G

is the gravitational constant,

m

is the mass of the particle,

M

is the mass of the Earth, and

r

is the distance from the center of the Earth to the particle.

Apply Newton's second law,

F = m a

, where

a

is the acceleration. Substituting the gravitational force, we get:

m a = \frac{G m M}{r^{2}}

. Cancel out

m

(the mass of the particle) to find the acceleration:

a = \frac{G M}{r^{2}}

Use the chain rule to relate acceleration to velocity and position:

a = \frac{d}{v} d t

and

v \frac{d}{r} d t

. Substituting, we get:

a = v \frac{d}{v} d r

. Replace

a

with the gravitational acceleration:

v \frac{d}{v} d r = \frac{G M}{r^{2}}

Integrate both sides to find the velocity. On the left, integrate with respect to

v

\int_{0}^{v} v d v

, and on the right, integrate with respect to

r

\int_{r_{E}}^{r} \frac{G M}{r^{2}} d r

. Solve these integrals to find the relationship between

v

and

r

After integration, use the limits of integration to evaluate the constants. The result will give the velocity

v

as a function of the initial height

r_{E}

and the Earth's radius. This velocity represents the speed of the particle just before it hits the Earth's surface.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration (F = ma). This principle is fundamental in analyzing the motion of the particle as it falls towards Earth, allowing us to relate the gravitational force to the particle's acceleration and ultimately its velocity.

Law of Universal Gravitation

The Law of Universal Gravitation, formulated by Isaac Newton, states that every mass attracts every other mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law is crucial for determining the gravitational force acting on the particle as it falls from a height above the Earth's surface.

Integration and the Chain Rule

Integration is a mathematical process used to find the total accumulation of a quantity, such as velocity from acceleration. The chain rule is a fundamental technique in calculus that allows us to differentiate composite functions. In this context, integrating the gravitational force with respect to the height will help us derive the velocity of the particle as it impacts the Earth.

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A particle is released at a height rE (radius of Earth) above the Earth’s surface. Determine its velocity when it hits the Earth. Ignore air resistance. [Hint: Use Newton’s second law, the law of universal gravitation, the chain rule, and integrate.]

Verified video answer for a similar problem:

Key Concepts

Newton's Second Law

Law of Universal Gravitation

Integration and the Chain Rule

A particle is released at a height r_E (radius of Earth) above the Earth’s surface. Determine its velocity when it hits the Earth. Ignore air resistance. [Hint: Use Newton’s second law, the law of universal gravitation, the chain rule, and integrate.]