Projectile with Linear Air Drag from a Moving Car
Problem
**Problem Statement** A car moves horizontally with a constant velocity of 20 m/s. From this moving car, a ball is thrown straight upward with an initial vertical velocity of 10 m/s relative to the car. The ball experiences two forces: gravity acting downward and air resistance proportional to its velocity. The drag force always acts opposite to the direction of motion. The ball has a mass of 0.2 kg, the linear drag coefficient is 0.1 kg/s, and gravitational acceleration is 9.8 m/s². At time t = 0, the ball is at position x = 0, y = 0 with initial velocities v_x = 20 m/s and v_y = 10 m/s. You must model the ball’s horizontal and vertical position and velocity over time using the given physical forces and determine whether the ball ever returns to the moving car. The car continues moving at constant velocity, so its position is x_c(t) = 20t. Your task is to compute the motion of the ball, find the time at which it lands (when y returns to 0), and check whether its horizontal position matches the car’s position at that moment. Use the equations of motion under linear drag: dv_x/dt = -(k/m) v_x dv_y/dt = -g -(k/m) v_y dx/dt = v_x dy/dt = v_y
Explanation
This visualization shows the 2D motion of a ball thrown upward from a horizontally moving car, including linear air resistance. The car moves at constant horizontal speed while the ball slows horizontally due to drag and accelerates downward due to gravity and drag. You can adjust physical parameters and see how the ball’s trajectory compares to the car’s path and whether it ever returns to the car (same x position when y = 0). The exact analytic solutions for linear drag are used, and the moment when the ball lands and how far it is from the car are highlighted.