Final answer:
To estimate the body's velocity and position at any time t > 0 in a resisting medium, solve the differential equation dv/dt = -kv. The body travels only a finite distance as it eventually comes to a stop. The distance traveled can be calculated by integrating the velocity function from t = 0 to t = infinity.
Step-by-step explanation:
The motion of a body moving through a resisting medium is described by the differential equation dv/dt = -kv, where v is the velocity and t is the time. To estimate the body's velocity and position at any time t > 0, we can solve this differential equation. Integrating both sides of the equation gives us v = v0 * exp(-kt), where v0 is the initial velocity. Substituting this expression for v into the equation dx/dt = v, we can solve for x(t) and find the position of the body at any time t > 0.
To determine that the body travels only a finite distance, we can look at the limit as t approaches infinity. As t approaches infinity, the exponential term exp(-kt) approaches 0, meaning that the velocity v approaches 0. This implies that the body eventually comes to a stop and does not continue to travel indefinitely. The distance traveled by the body can be calculated by integrating the velocity function from t = 0 to t = infinity.