Question

Consider an object falling vertically to Earth. Let the differential equation of motion be d²x /dt²m = mg - bv² where b is the damping coefficient (with units of kg/m).

Determine the terminal velocity (when the net force is zero) in terms of m, g and b?

Answer 1

To find the terminal velocity when an object falls toward Earth with air resistance, balance the gravitational and damping forces and solve for velocity to get v = sqrt(mg/b) where m is the mass, g is the acceleration due to gravity, and b is the damping coefficient.

Step-by-step explanation:

To determine the terminal velocity of an object falling to Earth with a damping coefficient, we set the net force to zero. The equation of motion given is d²x /dt²m = mg - bv² where m is the mass, g is the acceleration due to gravity, b is the damping coefficient, and v is the velocity. At terminal velocity, the acceleration is zero because the object is no longer accelerating and has reached a constant velocity; thus, the forces are balanced. Therefore, mg = bv².

To solve for terminal velocity v, we re-arrange the equation to v = sqrt(mg/b). By plugging in the values for m, g (typically 9.8 m/s²), and b, we can find the terminal velocity in terms of m, g and b.