Motion with Resistance Proportional to Velocity

Definition

This model describes the motion of an object through a resistive medium (like air or water) where the drag force is proportional to the object’s velocity. Based on Newton’s Second Law, the differential equation for velocity $v(t)$ is: $m \frac{dv}{dt} = -kv$

• How to read: “The mass m times the derivative of velocity with respect to time is equal to negative k times the velocity.”
• Meaning: Mass times acceleration equals drag force; drag opposes motion and scales linearly with velocity.

where $m$ is mass and $k$ is the resistance constant.

Why It Matters

In the real world, objects move through fluids (air, water) that push back. Ignoring velocity-proportional resistance leads to dangerously inaccurate predictions for parachutes, fuel efficiency, and terminal velocity. It is the difference between a safe landing and a fatal impact.

Core Concepts

• Exponential Decay: The velocity decreases exponentially over time: $v(t) = v_0 e^{-(k/m)t}$.

  • How to read: “The velocity at time t is equal to the initial velocity v zero times e raised to the power of negative k divided by m, times t.”
  • Meaning: Velocity decays exponentially; larger $k/m$ means faster slowdown.

• Terminal Distance: Even though the object technically moves forever as $t \to \infty$, the total distance traveled is finite and converges to $s_{\infty} = \frac{mv_0}{k}$.

  • How to read: “The terminal distance s infinity is equal to the mass m times the initial velocity v zero, divided by k.”
  • Meaning / when to use: Total coasting distance is finite despite never fully stopping—use for braking distance estimates.

• Deceleration Rate: The ratio $k/m$ determines how quickly the object slows down; heavier objects or those with less resistance coast further.

Connected Concepts

• Simple Harmonic Motion
• Tangent Problem and Velocity Problem
• Kepler’s Laws of Planetary Motion