Topic 2.9 Notes – Resistive Forces

1. What a Resistive Force Is

A resistive force depends on velocity and always points opposite the object’s motion.

For linear (low-speed) drag:

$$\vec F_{r} = -k \vec v$$

• $k > 0$ depends on the medium, shape, cross-sectional area.
• The minus sign guarantees the force opposes $\vec v$.
• If velocity flips direction, the drag force flips too.

Examples you’ll see:

• Air resistance at low speeds
• Viscous drag in fluids

Since $F_{r}$ depends on $v$, Newton’s Second Law $$\sum \vec F = m \frac{d\vec v}{dt}$$ becomes a differential equation, not a constant-acceleration situation. That’s the big shift here.

2. The Differential Equation for Motion with Linear Drag

Consider 1D motion with a constant force $F_{\text{const}}$ and drag:

$$m \frac{dv}{dt} = F_{\text{const}} - kv$$

This is a first-order linear differential equation.

Solving by Separation of Variables

Rearrange:

$$\frac{dv}{F_{\text{const}} - kv} = \frac{dt}{m}$$

Now integrate both sides using proper limits. If $v(0)=v_{0}$:

$$\int_{v_{0}}^{v(t)} \frac{dv}{F_{\text{const}} - kv} = \int_{0}^{t} \frac{dt}{m}$$

The left side gives a natural log. After algebra, you get an exponential in time. That exponential behavior is the signature of linear drag.

On a free-response question, most of the points come from:

• Writing Newton’s 2nd Law correctly with signs
• Separating variables cleanly
• Applying the initial condition correctly

Algebra mistakes usually happen when solving for $v(t)$ at the end.

3. Velocity, Acceleration, and Position Functions

Velocity as a Function of Time

The general solution is:

$$v(t) = v_{\text{terminal}} + (v_{0} - v_{\text{terminal}}) e^{-t/\tau}$$

Where:

• $v_{\text{terminal}} = \dfrac{F_{\text{const}}}{k}$
• $\tau = \dfrac{m}{k}$ (time constant)

What this means physically:

• Velocity approaches $v_{\text{terminal}}$ exponentially
• It never actually reaches it in finite time
• If $v_{0} > v_{t}$, it decreases toward $v_{t}$
• If $v_{0} < v_{t}$, it increases toward $v_{t}$

Time Constant $\tau = m/k$

This tells you how fast the system responds.

• After one $\tau$, velocity is about 63% of the way to terminal velocity.
• After about $3\tau$, it’s ~95% there.
• Larger mass → larger $\tau$ → slower approach.
• Larger $k$ → smaller $\tau$ → faster approach.

This shows up in multiple-choice as conceptual questions about “which object reaches terminal velocity faster?”

Acceleration as a Function of Time

Since $a = dv/dt$, acceleration is also exponential.

• Initially large (depending on forces and $v_{0}$)
• Decreases toward zero
• At terminal velocity, net force = 0 → $a = 0$

Position as a Function of Time

Position comes from integrating velocity:

$$x(t) = x_{0} + \int v(t)\, dt$$

You’ll get:

• Linear terms in $t$
• Exponential terms

Long-term behavior becomes approximately linear because velocity levels off to a constant.

No constant-acceleration kinematics applies here. Ever.

4. Terminal Velocity

Terminal velocity happens when net force is zero.

For a falling object with gravity downward and drag upward:

$$mg - kv_{\text{terminal}} = 0$$

$$v_{\text{terminal}} = \frac{mg}{k}$$

At this speed:

• Acceleration is zero
• Velocity is constant
• Motion continues at steady speed

Heavier object → larger $v_{t}$ (if $k$ is the same).
Larger drag constant → smaller $v_{t}$.

Here’s what the velocity graph looks like for a falling object.

Velocity vs. time approaching terminal velocity

The curve rises quickly at first, then levels off as it approaches $v_{t}$. Notice the horizontal asymptote. That’s what the AP loves to test. They often ask what happens “as $t \to \infty$.”

5. How to Analyze Resistive-Force Problems

Falling Object

1. Choose a positive direction.
2. Write Newton’s 2nd Law with correct signs.
3. Solve for $v(t)$.
4. Use:
   • $v=0$ for turning points
   • Fraction of $v_{t}$ for time-to-percentage questions
5. Integrate if you need position.

Object Thrown Upward

Be careful with direction:

• On the way up: gravity and drag both downward.
• On the way down: gravity downward, drag upward.

Students often mess up the sign of drag when velocity changes direction. The safest move is to write drag as $-k v$ and let the sign of $v$ handle everything.

Expect:

• Lower maximum height than no-drag case
• Exponential velocity behavior
• Asymptotic approach to downward terminal velocity

Everything flows from Newton’s Second Law plus calculus.