Problem 12-34: Engineering Mechanics- Dynamics (Hibbeler 14th Edition)

Question:
Accounting for the variation of gravitational acceleration a with respect to altitude y, derive an equation that relates the velocity of a freely falling particle to its altitude. Assume the particle is released from rest at altitude y-naught above Earth's surface. With what velocity does the particle strike Earth if it is released from rest at y-naught equals 500 kilometers? Use g equals 9.81 meters per second squared and R equals 6,371 kilometers.

Velocity–Altitude Equation for Freefall with Variable Gravity

This lesson derives a velocity–altitude relationship for a freely falling particle using Newton's inverse-square law of gravitation and the kinematic chain rule. The derivation integrates a separable ODE, producing a closed-form equation that is then evaluated numerically for a particle released from rest at an altitude of 500 km above Earth's surface.


In this lesson,