A rocket fired vertically up from the north pole achieves a...

A rocket fired vertically up from the north pole achieves a velocity of $27000 \mathrm{km} / \mathrm{h}$ at an altitude of $350 \mathrm{km}$ when its fuel is exhausted. Calculate the additional vertical height $h$ reached by the rocket before it starts its descent back to the earth. The coasting phase of its flight occurs above the atmosphere. Consult Fig. $1 / 1$ in choosing the appropriate value of gravitational acceleration and use the mean radius of the earth from Table D/2. (Note: Launching from the earth's pole avoids considering the effect of the earth's rotation.)

A rocket fired vertically up from the north pole achieves a velocity of $27000 \mathrm{km} / \mathrm{h}$ at an altitude of $350 \mathrm{km}$ when its fuel is exhausted. Calculate the additional vertical height $h$ reached by the rocket before it starts its descent back to the earth. The coasting phase of its flight occurs above the atmosphere. Consult Fig. $1 / 1$ in choosing the appropriate value of gravitational acceleration and use the mean radius of the earth from Table D/2. (Note:
Launching from the earth's pole avoids considering the effect of the earth's rotation.)

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Key Concepts

Variation of Gravitational Acceleration with Altitude

Gravitational acceleration decreases with increasing distance from the Earth’s center. When analyzing the rocket’s coasting phase above the atmosphere, it is important to use the appropriate value of gravitational acceleration that accounts for the altitude, ensuring accurate calculations of the potential energy change and hence the additional altitude gained during coasting.

Conservation of Mechanical Energy

This principle states that in an isolated system where only conservative forces act, the sum of kinetic and potential energy remains constant. In the context of a vertically launched rocket with fuel exhaustion, the kinetic energy at the beginning of the coast phase is converted into gravitational potential energy as the rocket ascends until its upward speed reduces to zero.

Gravitational Potential Energy

Gravitational potential energy quantifies the energy a mass has due to its position in a gravitational field. For objects at significant altitudes relative to Earth’s radius, this energy is calculated using formulas that consider the radial distance from Earth’s center, highlighting that the increase in altitude results in a decrease in the gravitational force experienced by the object.

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