   Chapter 7, Problem 39P

Chapter
Section
Textbook Problem

A projectile is fired straight upward from the Earth’s surface at the South Pole with an initial speed equal to one third the escape speed, (a) Ignoring air resistance, determine how far from the center of the Earth the projectile travels before stop-ping momentarily, (b) What is the altitude of the projectile at this instant?

(a)

To determine
The distance that the projectile travels from the centre of the earth before it stops momentarily.

Explanation

Given Info:

The air resistance of the projectile is ignored. The speed of the projectile is one third the escape speed and radius of the earth is 6.38×106m .

Explanation:

Since, the air resistance of the projectile is ignored; the only force that is acting on the projectile is the gravitational force. Gravitational force is a conservative force. Thus, the total energy of the projectile is a constant.

The speed of the projectile is,

v=vesc3       (I)

• vesc is the escape velocity

The escape velocity is,

vesc=2GMR       (II)

• G is the gravitational constant
• M is the mass of Earth
• R is the radius of Earth

Substitute equation (II) in (I),

v=2GMR3       (III)

The total energy of the projectile is,

E=KE+PEg

E=12mv2+(GMmR)       (IV)

• m is the mass of projectile

On substituting equation (III) in (IV) and rearranging,

Formula to calculate the total energy of the projectile is,

E=89GMmR       (V)

Thus, the total energy of the projectile is, E=89GMmR

(b)

To determine
The altitude of the projectile when the projectile reaches the maximum height.

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