Resistive Forces
A resistive force depends on velocity and points opposite the velocity. For the course model F⃗_r=−k v⃗, Newton’s second law becomes a first-order differential equation for velocity; separation of variables produces exponential approach to terminal velocity.
Topic sequence · 9 of 10
Learning objectives
Objective 1
Model linear resistive force with F⃗_r=−k v⃗ and correct direction.
Objective 2
Set up and solve the corresponding separable differential equation for velocity.
Objective 3
Interpret terminal velocity, time constant, exponential graphs, and asymptotic behavior.
- Newton’s second law as a differential equation from Topic 2.5.
- Exponential functions, integration, logarithms, and separation of variables.
- Sign conventions for one-dimensional motion.
Definitions first
Representations you must read and create
Representation 1
F_r versus v graph: straight line through origin with negative slope −k in one dimension.
Representation 2
Velocity-time exponential approaching v_t.
Representation 3
Acceleration-time exponential approaching zero as the net force vanishes.
Core relationships and calculus
Variables: k kg/s; v m/s
Use: linear resistive-force model
Variables: downward chosen positive
Use: falling object with gravity and linear drag
Variables: same sign as constant driving force
Use: terminal speed for gravity + linear drag
Variables: τ=m/k
Use: velocity solution for constant drive + linear drag
Variables: derivative of exponential velocity
Use: time-dependent acceleration
From m dv/dt=mg−kv, write dv/(v_t−v)=(k/m)dt and integrate with limits v0 to v and 0 to t. Exponentiating the resulting logarithm gives the exponential solution. Position follows from a second integration of v(t), while acceleration follows from differentiation.
Separate variables: dv/(g−(k/m)v)=dt. Integrating yields a logarithm; applying the initial condition fixes the integration constant. Rewriting with τ=m/k and v_t=mg/k makes the physical scales obvious.
The current quantitative resistive-force model is linear drag F_r=−kv. Do not substitute quadratic drag or assume constant acceleration when velocity-dependent resistance is present unless a problem explicitly supplies a different model.
Physical examples
Example 1
A falling object released from rest initially has zero drag, so its initial acceleration is g; drag grows as speed increases.
Example 2
At terminal velocity, the object still moves but acceleration is zero because upward drag balances downward weight.
Example 3
A larger k lowers terminal speed and shortens τ=m/k, so the object approaches a smaller limiting speed more quickly.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Choose a positive direction first; then the drag sign follows automatically from “opposite v.”
Reasoning pattern 2
Before integrating, identify the terminal condition by setting dv/dt=0; this supplies a powerful check on the eventual solution.
Reasoning pattern 3
Interpret τ=m/k and the exponential asymptote physically, not just algebraically.
Deeper calculus and limiting-case check
For release from rest under downward gravity and linear drag, v/v_t=1−e^(−t/τ). This nondimensional form shows that all such motions share the same curve when time is measured in units of τ and velocity in units of v_t.
Model-selection workflow: decide what the problem is before choosing an equation
For Resistive Forces, the fastest reliable route is not equation hunting. Start by identifying the system and interaction, encode it with the free-body diagram, choose the Newton-law or momentum-rate equation, and then test the inertial-frame and force-model assumption. This sequence prevents a mathematically correct manipulation from being attached to the wrong physical model.
Define the physical object or system
Name what is being modeled, what is outside the system, the coordinate or orientation convention, and the interval or geometry over which the statement applies. Treat resistive force and linear drag coefficient as physical quantities first and symbols second.
Choose a representation
Build F_r versus v graph: straight line through origin with negative slope −k in one dimension.. If the prompt supplies Velocity-time exponential approaching v_t., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is F⃗_r = −k v⃗. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The current quantitative resistive-force model is linear drag F_r=−kv. Do not substitute quadratic drag or assume constant acceleration when velocity-dependent resistance is present unless a problem explicitly supplies a different model. After solving, test units, sign or direction, zero/large-parameter limits, and whether the answer remains consistent with the original representation.
Representation → equation → calculus → physics chain
| Stage | What to write or inspect | What proves the step is physically valid |
|---|---|---|
| Representation | F_r versus v graph: straight line through origin with negative slope −k in one dimension. | Every arrow, sign, axis, node, surface, path, or reference has an explicit meaning. The diagram must contain enough information to determine the relevant components or constraints. |
| Primary model | F⃗_r = −k v⃗ | linear resistive-force model. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | m dv/dt = mg − kv | falling object with gravity and linear drag. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | From m dv/dt=mg−kv, write dv/(v_t−v)=(k/m)dt and integrate with limits v0 to v and 0 to t. Exponentiating the resulting logarithm gives the exponential solution. Position follows from a second integration of v(t), while acceleration follows from differentiation. | State what the derivative, integral, differential equation, or limiting process represents physically; do not treat calculus notation as an algebraic decoration. |
| Physical check | Return to the original words and representation. | The final sign, direction, units, endpoint behavior, and functional dependence must all agree. A correct-looking expression that violates one of these checks needs repair. |
Advanced synthesis: how to handle an unfamiliar version of this topic
Take downward as positive for a 2 kg object with k=4 kg/s. Then τ=m/k=0.50 s and v_t=mg/k≈4.90 m/s. Released from rest, v(t)=4.90[1−e^(−t/0.50)] m/s. At t=0.50 s, v≈3.10 m/s, about 63.2% of terminal speed. To deepen this example, change one assumption or parameter, predict the effect before recomputing, and then use the symbolic dependence to verify the prediction.
FRQ-quality response architecture for Topic 2.9
Mathematical routines
Start from the governing law or definition, keep the derivation symbolic, identify the condition that permits any simplification, and finish with units plus a limiting-case check. For this topic, the relationship F⃗_r = −k v⃗ should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between F_r versus v graph: straight line through origin with negative slope −k in one dimension. and Velocity-time exponential approaching v_t.. Explain what a slope, area, direction, curvature, intercept, node value, field orientation, or other feature means instead of merely redrawing it.
Experimental design and analysis
Turn the topic into a measurable relationship. State the independent and dependent variables, important controls, repeated/ranged trials, and the graph or transformed graph whose slope/intercept would test the model or extract a physical parameter.
Qualitative/quantitative translation
Make the comparison or direction claim first, justify it through the model, derive the symbolic dependence independently, and then reconcile the derivation with the original qualitative claim.
Do not let an equation stand in for an explanation. A complete response names what changes, identifies the governing physical relationship, follows the causal chain, and states the requested outcome explicitly.
Experiment and data-analysis upgrade
Use a falling coffee filter or a simulation designed for approximately linear drag. Record v(t), estimate v_t from the late-time plateau, and test whether ln|v_t−v| versus t is approximately linear with slope −1/τ.
Operationalize the variables
Say exactly what instrument or observable supplies each measured quantity. Distinguish a directly measured variable from a quantity calculated later from those measurements.
Predict the graph before collecting data
Use the governing relationship to decide what should be plotted. If the raw relationship is nonlinear, choose a transformation that produces a line whose slope or intercept has a defined physical meaning.
Control and repeat for a reason
Hold variables fixed when they would change the predicted relationship. Repeat trials to estimate scatter; vary the independent quantity over a useful range to test functional dependence. These solve different experimental problems.
Evaluate the model, not only the number
Compare the best-fit result with the theoretical prediction, inspect residual or endpoint behavior when appropriate, and identify whether a discrepancy is random scatter, instrument resolution, or a systematic violation of an ideal-model assumption.
Error-repair matrix
| Failure pattern | Repair move | Self-check |
|---|---|---|
| Treating drag as constant even though the model explicitly depends on velocity. | Return to the free-body diagram, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Pointing drag upward in every problem instead of opposite the instantaneous velocity. | Return to the Newton-law or momentum-rate equation, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
| Thinking terminal velocity means zero velocity; it means zero acceleration with nonzero constant velocity. | Return to the free-body diagram, state the missing condition or convention explicitly, and rebuild the step before doing arithmetic. | Can you explain why the corrected step must have its sign, direction, units, or functional form without looking at the final answer? |
Spiral review: make this topic transferable
A topic is exam-ready only when it can be recognized inside a mixed problem. Use the connections below as transfer tests rather than as optional reading.
Transfer 1
Topic 2.5 supplies the differential-equation framework. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 2.6 provides the constant gravitational driving force in the standard falling model. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Unit 3 later accounts for energy transferred out of the mechanical system by resistive interactions. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Formula-condition retrieval
From memory, write F⃗_r = −k v⃗ and one condition that must be true before using it. Then state one situation where a more general derivative, integral, conservation law, or field/force model is required.
Topic 2.9 mastery ladder
Interactive linear-drag solution
Downward is positive. Evaluate τ=m/k, v_t=mg/k, v(t), and a(t) for linear drag.
Experimental connection
Use a falling coffee filter or a simulation designed for approximately linear drag. Record v(t), estimate v_t from the late-time plateau, and test whether ln|v_t−v| versus t is approximately linear with slope −1/τ.
Connections across Physics C
Connection 1
Topic 2.5 supplies the differential-equation framework.
Connection 2
Topic 2.6 provides the constant gravitational driving force in the standard falling model.
Connection 3
Unit 3 later accounts for energy transferred out of the mechanical system by resistive interactions.
Related lessons and resources
Use these links when the current model depends on a prerequisite or becomes the foundation for the next model.
Common mistakes
Avoid this
Treating drag as constant even though the model explicitly depends on velocity.
Avoid this
Pointing drag upward in every problem instead of opposite the instantaneous velocity.
Avoid this
Thinking terminal velocity means zero velocity; it means zero acceleration with nonzero constant velocity.
Mastery check
- Define the main quantity without relying on an equation.
- Choose the correct representation and sign convention.
- Explain why the calculus relationship matches the physics.
- Design or interpret an experiment that could test the model.
Frequently asked questions
Why is the approach exponential?
The remaining difference v_t−v controls the net force, so the rate of change of that difference is proportional to the difference itself.
What does τ=m/k mean physically?
It sets the time scale for approaching terminal behavior; after one τ, a released object has completed about 63% of the change toward v_t.
Is quadratic drag part of this model?
The current core treatment emphasizes a velocity-dependent resistive force such as the linear model F_r=−kv; other drag laws require different differential equations.