Introduction to Classical Mechanics · Problem 3.43
Using F = ma — Morin
Physics — Mechanics · projectile_motion
Problem
Increasing gravity * At on the planet Gravitus Increasicus, a projectile is fired with speed at an angle above the horizontal. This planet is a strange one, in that the acceleration due to gravity increases linearly with time, starting with a value of zero when the projectile is fired. In other words, , where is a given constant. What horizontal distance does the projectile travel? What should be to maximize this distance?Intuitive explanation
The key is recognizing that with time-dependent gravity , you can't use the standard kinematic equations—instead, you must integrate the acceleration to find velocity, then integrate velocity to find position. The vertical motion will be particularly tricky: gravity starts at zero and grows, so the projectile rises longer than it would on Earth before falling back. Set up the equations of motion separately for and , integrate twice using the initial conditions, then find when the projectile returns to ground level to determine range.
Intuitive explanation
The key is recognizing that with time-dependent gravity , you can't use the standard kinematic equations—instead, you must integrate the acceleration to find velocity, then integrate velocity to find position. The vertical motion will be particularly tricky: gravity starts at zero and grows, so the projectile rises longer than it would on Earth before falling back. Set up the equations of motion separately for and , integrate twice using the initial conditions, then find when the projectile returns to ground level to determine range.