Displacement, Velocity, and Acceleration
Physics C defines velocity and acceleration through derivatives and recovers displacement or velocity through definite integrals. Average and instantaneous quantities must be kept distinct.
Topic sequence · 2 of 5
Learning objectives
Objective 1
Distinguish position, displacement, average velocity, instantaneous velocity, average acceleration, and instantaneous acceleration.
Objective 2
Use limits, derivatives, and definite integrals to translate between position, velocity, and acceleration functions.
Objective 3
Interpret signs and initial conditions physically rather than treating derivatives as purely algebraic operations.
- Topic 1.1 vector components and signs.
- Derivatives, definite integrals, and function notation.
Definitions first
Representations you must read and create
Representation 1
Position, velocity, and acceleration functions with units attached to every coefficient.
Representation 2
Average-rate secant slopes approaching instantaneous tangent slopes as Δt→0.
Representation 3
Matched x(t), v(t), and a(t) graphs whose slopes and signed areas agree with the calculus relationships.
Core relationships and calculus
Variables: x position (m); t time (s)
Use: instantaneous velocity
Variables: a acceleration (m/s²)
Use: instantaneous acceleration
Variables: signed area under v–t graph
Use: displacement from velocity
Variables: signed area under a–t graph
Use: velocity change from acceleration
Variables: finite interval Δt; displacement Δx
Use: average velocity
Variables: finite interval Δt; velocity change Δv
Use: average acceleration
Use differentiation when the model gives position and asks for rates. Use definite integration when the model gives a rate and asks for accumulated change. Keep integration constants only when reconstructing a full function from an indefinite integral.
Average velocity is Δx/Δt and average acceleration is Δv/Δt. Taking the limit as Δt approaches zero produces v=dx/dt and a=dv/dt. Reversing those operations with definite integrals gives displacement from velocity and change in velocity from acceleration. Initial conditions are required when an integral is used to reconstruct a complete function.
The object model ignores size, shape, and internal configuration when those details are not needed. Use derivatives and integrals for instantaneous or time-varying motion; constant-acceleration shortcuts belong only to the conditions stated in Topic 1.3.
Physical examples
Example 1
If x(t)=2t³−5t, then v(t)=6t²−5 and a(t)=12t; the signs of v and a can change independently.
Example 2
A positive area and negative area on a velocity-time graph can cancel, producing zero displacement even when total distance is nonzero.
Example 3
Acceleration can be negative while an object speeds up when velocity is also negative.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Separate average quantities over a finite interval from instantaneous quantities at one time.
Reasoning pattern 2
If a function is provided, differentiate or integrate first and only then evaluate at the requested time or interval.
Reasoning pattern 3
Use signs of v and a together to decide whether an object is moving in +/− direction and speeding up or slowing down.
Deeper calculus and limiting-case check
For motion along one axis, a can also be written a=(dv/dx)(dx/dt)=v dv/dx when velocity is known as a function of position. That chain-rule form becomes useful when force depends on position and time is not the most convenient independent variable.
Model-selection workflow: decide what the problem is before choosing an equation
For Displacement, Velocity, and Acceleration, the fastest reliable route is not equation hunting. Start by identifying the state variable, encode it with the vector component and graph, choose the derivative/integral relationship, and then test the constant-versus-variable motion 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 position and displacement as physical quantities first and symbols second.
Choose a representation
Build Position, velocity, and acceleration functions with units attached to every coefficient.. If the prompt supplies Average-rate secant slopes approaching instantaneous tangent slopes as Δt→0., translate it into the same sign, axis, orientation, or reference convention before combining information.
Write the governing relationship
A useful starting relationship is v_x=dx/dt. State what the variables mean and why this relation—not merely a familiar formula—matches the physical situation.
Check conditions and limits
The object model ignores size, shape, and internal configuration when those details are not needed. Use derivatives and integrals for instantaneous or time-varying motion; constant-acceleration shortcuts belong only to the conditions stated in Topic 1.3. 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 | Position, velocity, and acceleration functions with units attached to every coefficient. | 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 | v_x=dx/dt | instantaneous velocity. Preserve the variables and units long enough to see dependence before inserting numerical values. |
| Second relationship | a_x=dv_x/dt=d²x/dt² | instantaneous acceleration. Combine relationships only after confirming they describe the same system, interval, frame, or geometry. |
| Calculus operation | Use differentiation when the model gives position and asks for rates. Use definite integration when the model gives a rate and asks for accumulated change. Keep integration constants only when reconstructing a full function from an indefinite integral. | 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
For x(t)=t³−6t²+9t (m), v=3t²−12t+9 and a=6t−12. At t=1 s, v=0 while a=-6 m/s², so the object is instantaneously at rest but not in equilibrium; its velocity changes immediately after that instant. 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 1.2
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 v_x=dx/dt should appear only when its physical conditions have been established.
Translation between representations
Move deliberately between Position, velocity, and acceleration functions with units attached to every coefficient. and Average-rate secant slopes approaching instantaneous tangent slopes as Δt→0.. 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
Track a cart with a motion sensor. Compare numerical slopes of x(t) with measured v(t), and numerical slopes of v(t) with a(t). Quantify how sampling interval affects derivative noise.
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 |
|---|---|---|
| Calling displacement distance. | Return to the vector component and graph, 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? |
| Assuming v=0 implies a=0. | Return to the derivative/integral relationship, 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? |
| Using constant-acceleration equations when a(t) is not constant. | Return to the vector component and graph, 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 1.3 turns these derivative and integral definitions into graph-reading rules. Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 2
Topic 1.5 applies the same derivatives independently to x(t), y(t), and z(t). Re-solve one problem from that connected topic and identify the exact step where the present model enters or stops applying.
Transfer 3
Topic 2.5 turns a=d²x/dt² into the differential equation generated by a net force. 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 v_x=dx/dt 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 1.2 mastery ladder
Interactive derivative check
For x(t)=at²+bt+c, change the coefficients and time.
Experimental connection
Track a cart with a motion sensor. Compare numerical slopes of x(t) with measured v(t), and numerical slopes of v(t) with a(t). Quantify how sampling interval affects derivative noise.
Connections across Physics C
Connection 1
Topic 1.3 turns these derivative and integral definitions into graph-reading rules.
Connection 2
Topic 1.5 applies the same derivatives independently to x(t), y(t), and z(t).
Connection 3
Topic 2.5 turns a=d²x/dt² into the differential equation generated by a net force.
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
Calling displacement distance.
Avoid this
Assuming v=0 implies a=0.
Avoid this
Using constant-acceleration equations when a(t) is not constant.
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
Does v=0 mean the object is in equilibrium?
No. Velocity may be zero at an instant while acceleration is nonzero, as at a turning point.
When should I use an integral instead of a constant-acceleration equation?
Use an integral whenever the given rate varies or when the problem supplies a general function rather than a constant acceleration.
Why can displacement be zero while distance is not?
Displacement is the signed change in position; motion in opposite directions can cancel in the net displacement.
Topic sequence · 2 of 5
Individually reviewed Mechanics practice: displacement, velocity and acceleration
These questions replace abstract cues with precise position functions, reference frames and answer explanations. They are supplementary to the original question bank.
M-1.2-Q01 — A cart is located at x=+7 m relative to an origin at x=0. If the coordinate origin moves +3 m while the cart stays still, what is its new x-coordinate?
- +10 m
- +4 m
- −4 m
- +7 m
Correct answer: B. Coordinates depend on the reference origin. The new coordinate is x′=7−3=+4 m. The cart has not physically moved, and a change of coordinates is not itself a displacement.
M-1.2-Q02 — The position of a particle is x(t)=2t²−4t metres for t in seconds. What is its displacement between t=0 and t=3 s?
- −6 m
- +18 m
- +6 m
- +12 m
Correct answer: C. Displacement is final minus initial position. x(3)=2(9)−4(3)=+6 m and x(0)=0, so Δx=+6 m. Do not substitute velocity into the displacement formula.
M-1.2-Q03 — A particle follows x(t)=t³−2t² metres, with time in seconds. What is its instantaneous velocity at t=2.0 s?
- 0 m/s
- +2 m/s
- +8 m/s
- +4 m/s
Correct answer: D. Differentiate x with respect to time: v(t)=3t²−4t. Therefore v(2)=3(4)−4(2)=+4 m/s. Position x(2)=0 m does not imply velocity zero.
Original free response: variable velocity, displacement and total distance
Scenario. A particle travels along the x-axis for 0 ≤ t ≤ 3 s with position x(t)=2t³−9t²+12t metres. The positive x-axis points right. The function is smooth over the entire time interval.
- (a), 2 practice points: Derive velocity and acceleration functions. Solution:
v(t)=6t²−18t+12=6(t−1)(t−2)m/s, anda(t)=12t−18m/s². - (b), 3 points: Determine intervals of rightward versus leftward motion and the instants of a change in direction. Solution: v>0 on 0 ≤ t<1 and 2<t ≤ 3; v<0 on 1<t<2. Velocity vanishes and changes sign at t=1 s and t=2 s.
- (c), 3 points: Find displacement and total distance during 0–3 s. Solution: x(0)=0, x(1)=5, x(2)=4 and x(3)=9 m. Displacement is 9−0=9 m right; total distance is |5−0|+|4−5|+|9−4|=11 m.
- (d), 2 points: Find average velocity and average speed over the interval. Solution: Average velocity =9 m/3 s=+3.00 m/s; average speed =11 m/3 s≈3.67 m/s.
Self-check rubric: 2 points for accurate derivatives and units; 3 for correct roots and sign intervals; 3 for the displacement-versus-distance calculation using turning points; 2 for distinguishing average vector velocity from average scalar speed. This is original practice, not an official scoring guideline.