Review AP Physics C: Mechanics Unit 1 to build the foundation for every motion problem in the course. This unit covers vectors, calculus-based definitions of velocity and acceleration, kinematic equations, reference frames, and two-dimensional motion including projectile analysis.
Use the topic guides, practice questions, and FRQ practice available for this unit to work through each concept before moving to Unit 2.
Kinematics is the description of motion without asking why it happens. AP Physics C: Mechanics treats this topic with calculus, so velocity is the derivative of position and acceleration is the derivative of velocity. Reversing those relationships with integration lets you reconstruct position or velocity from a graph or function.
Every kinematic quantity except distance and speed is a vector. Expressing vectors in unit vector notation (î, ĵ, k̂) lets you handle components algebraically and is required for two- and three-dimensional problems throughout the course.
Instantaneous velocity is dx/dt and instantaneous acceleration is dv/dt. Integrating acceleration gives change in velocity; integrating velocity gives displacement. On a graph, slope gives the derivative and area under the curve gives the integral.
Separating motion into perpendicular x and y components lets you apply one-dimensional kinematic equations in each direction independently. Time is the shared variable that links the two components, which is the key to solving projectile problems.
The core skill of Unit 1 is translating between representations of motion: functions, graphs, diagrams, and equations. Whether you are reading the slope of a position-time graph, integrating an acceleration function, or decomposing a launch velocity into components, you are applying the same underlying relationships between position, velocity, and acceleration.
Distinguish scalar quantities (magnitude only) from vector quantities (magnitude and direction). Express vectors in unit vector notation using î, ĵ, and k̂, and find resultant vectors by adding components in each direction.
Define displacement, average velocity, and average acceleration. Use derivatives to find instantaneous velocity (dx/dt) and instantaneous acceleration (dv/dt), and use integration to recover displacement or velocity from a function or graph.
Apply the three constant-acceleration kinematic equations to one-dimensional problems. Interpret position-time, velocity-time, and acceleration-time graphs using slopes and areas under curves.
Identify inertial reference frames and convert velocity measurements between frames using vector addition. Recognize that acceleration is the same in all inertial frames.
Decompose two-dimensional motion into independent x and y components. Analyze projectile motion by applying constant-acceleration equations horizontally and vertically, using time as the shared variable.
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Review Motion in Two or Three Dimensions with attention to how the concept appears in AP-style source and evidence questions.
Review Displacement, Velocity, and Acceleration with attention to how the concept appears in AP-style source and evidence questions.
A scalar is fully described by magnitude alone, such as distance, speed, or mass. A vector requires both magnitude and direction, such as displacement, velocity, acceleration, or force. Vectors are written in unit vector notation as the sum of components along each axis: r = A î + B ĵ + C k̂. The magnitude of a vector is found using the Pythagorean theorem across its components. A resultant vector is found by adding components in each direction separately.
| Quantity | Type | Example |
|---|---|---|
| Distance | Scalar | 5 m traveled along a path |
| Displacement | Vector | 3 î + 4 ĵ m |
| Speed | Scalar | 10 m/s |
| Velocity | Vector | 10 î m/s |
| Acceleration | Vector | -9.8 ĵ m/s^2 |
Displacement is the change in position: delta x = x - x0. Average velocity is displacement divided by elapsed time; average acceleration is change in velocity divided by elapsed time. As the time interval shrinks toward zero, these averages become instantaneous values. Instantaneous velocity is v = dr/dt and instantaneous acceleration is a = dv/dt. Integrating acceleration over time gives change in velocity; integrating velocity gives displacement. An object is accelerating whenever its velocity changes in magnitude, direction, or both.
| Quantity | Average form | Instantaneous form |
|---|---|---|
| Velocity | delta x / delta t | dx/dt |
| Acceleration | delta v / delta t | dv/dt |
Motion can be represented with graphs, equations, motion diagrams, and written descriptions. For constant acceleration in one dimension, three kinematic equations apply: v = v0 + at; x = x0 + v0 t + (1/2)a t^2; v^2 = v0^2 + 2a(x - x0). On a position-time graph, the slope at any point equals instantaneous velocity. On a velocity-time graph, the slope equals acceleration and the area under the curve equals displacement. Near Earth's surface, the gravitational acceleration is approximately 10 m/s^2 downward.
| Graph type | Slope represents | Area represents |
|---|---|---|
| Position vs. time | Instantaneous velocity | Not directly used |
| Velocity vs. time | Instantaneous acceleration | Displacement |
| Acceleration vs. time | Rate of change of acceleration | Change in velocity |
A reference frame is the coordinate system from which an observer measures motion. The same object can have different positions and velocities depending on the observer's frame. To convert between inertial frames, add or subtract velocity vectors: v_AC = v_AB + v_BC. Acceleration is the same in all inertial reference frames because the frames move at constant velocity relative to each other. Classic problems include a boat crossing a river with a current or a plane flying in wind.
Two-dimensional motion is analyzed by separating the position vector into x and y components and applying one-dimensional kinematic equations independently in each direction. The components share the same time variable, which links them. Projectile motion is the key special case: horizontal acceleration is zero (constant vx) and vertical acceleration is -g. The range, time of flight, and maximum height are derived from the component equations. Changing motion in one direction does not affect the perpendicular direction.
| Quantity | Horizontal (x) | Vertical (y) |
|---|---|---|
| Acceleration | 0 | -g (approx. -10 m/s^2) |
| Velocity | vx = v0 cos(theta), constant | vy = v0 sin(theta) - gt |
| Position | x = v0 cos(theta) t | y = v0 sin(theta) t - (1/2)g t^2 |
Try AP-style multiple-choice questions and written prompts after you review the notes.
2. A drone flies horizontally at a constant velocity of 15 m/s relative to the ground at an altitude of 80 m above level ground, as shown in Figure 1. At time t = 0 s, the drone releases a package. Air resistance is negligible.
Figure 1. Drone moving horizontally at constant speed releases a package from an altitude of 80 m above level ground at t = 0 s.
Figure 2. Velocity-component axes for the package as observed by (left) a ground observer and (right) a drone observer at t = 0 s, 2 s, and 4 s.
On the axes provided in Figure 2, draw velocity component vectors for the package at t = 0 s, t = 2 s, and t = 4 s as observed by both the ground observer and the drone observer. The length of each vector should be proportional to the magnitude of that velocity component. If a velocity component is zero, indicate this with a dot on the origin of the appropriate axes.
Derive an expression for the horizontal distance d traveled by the package from the moment of release until it hits the ground, as measured by the ground observer. Express your answer in terms of the drone's horizontal speed , the initial altitude , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The package lands on the ground at a horizontal distance from the release point.
Figure 3. x–y coordinate grid for sketching the package’s trajectory as seen by the ground observer.
On the coordinate grid in Figure 3, sketch the trajectory of the package from release until it hits the ground as observed by the ground observer. The trajectory should begin at the correct initial position and end at the correct final position. The trajectory of the package as observed by the ground observer can be represented on an x-y coordinate system.
Describe how one feature of the trajectory on the x-y coordinate grid would differ for the second package compared to the first package, as observed by the ground observer. Explicitly state whether the feature increases or decreases. A second identical package is released from the same drone when it is at the same altitude of 80 m but is now traveling at a horizontal speed of 20 m/s instead of 15 m/s.
Justify your answer using physics principles.
4. A student stands on a platform that moves horizontally to the right with constant velocity . At time , the student throws a ball upward with an initial speed relative to the platform at an angle above the horizontal, as shown in Figure 1. A stationary observer stands on the ground watching the ball's motion.
Figure 1. Ball thrown from a rightward-moving platform: velocities relative to the platform and the ground.
The horizontal component of the ball's velocity as measured by the student on the platform is , and the horizontal component of the ball's velocity as measured by the stationary observer on the ground is .
Indicate whether is greater than, less than, or equal to by writing one of the following.
Justify your answer using qualitative reasoning beyond referencing equations.
Derive an expression for the horizontal distance traveled by the object before it hits the ground. Express your answer in terms of , , , , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider a general case where an object is thrown with initial velocity components and (relative to the ground) from a height above the ground.
Figure 2. Vertical velocity component v_y versus time t (blank axes for student sketch).
Indicate whether the time the ball remains in the air in this new scenario is greater than, less than, or equal to the time the ball remains in the air in the original scenario. Use Figure 2 to sketch the vertical velocity component as a function of time for this new scenario. In a different scenario, the platform moves with the same speed to the right, but the student now throws the ball with the same initial speed relative to the platform at an angle backward (toward the left) and upward. The ball is released from a height of above the ground in both the original scenario and this new scenario.
Briefly justify your answer.
3. A cart of mass kg is placed on a track that is inclined at an angle above the horizontal, as shown in Figure 1. The cart is equipped with a low-friction wheel system such that friction between the cart and the track is negligible. A motion sensor is positioned at the bottom of the track to measure the velocity of the cart as it moves down the incline.
Figure 1. Cart on a frictionless inclined track with motion sensor and incline-aligned coordinate axes.
Students are asked to experimentally determine the acceleration due to gravity using a linear graph. To determine , the students are permitted to use measurements from only the motion sensor, meterstick, and protractor.
Describe an experimental procedure using the described setup to collect data that would allow the students to determine an experimental value of using a linear graph. Include any steps necessary to reduce experimental uncertainty.
Describe how the data collected in part A could be graphed and how that graph would be analyzed to determine the value of .
Figure 2. Velocity v versus time t showing constant positive acceleration (straight-line v–t relationship).
Figure 3. Adjustable-angle incline showing three discrete θ configurations about a fixed bottom pivot.
Figure 4. Blank grid for plotting data
(degrees) | (m/s²) |
|---|---|
10.0 | 1.68 |
15.0 | 2.51 |
20.0 | 3.38 |
25.0 | 4.18 |
30.0 | 4.95 |
In a different experiment, students set the track at various angles and measure the acceleration of the cart down the incline for each angle using the motion sensor. The adjustable setup is shown in Figure 3. The students' measurements are shown in Table 1.
Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine .
Vertical axis: Horizontal axis:
On the grid provided in Figure 4, create a graph of the quantities indicated in part C(i).
Use Table 2 to record the measured or calculated quantities that you will plot.
Clearly label the axes, including units as appropriate.
Plot the points you recorded in Table 2.
Draw a best-fit line to the data graphed in part C(ii).
Using the best-fit line that you drew in part C(iii), calculate an experimental value for . Using the best-fit line from part C(iii), a student determines that the slope of the line is m/s².
| Term | Definition |
|---|---|
| scalar | A physical quantity described by magnitude only, without direction; examples include distance and speed. |
| position vector | A vector r that specifies the location of a point relative to the origin of a coordinate system, written as r = x î + y ĵ + z k̂. |
| vector addition | Combining two or more vectors by adding their components in each direction to produce a resultant vector. |
| object model | A simplifying model that treats an object as a point particle, ignoring size and shape, when those details do not affect the motion analysis. |
| Kinematic equations | Three equations relating position, velocity, acceleration, and time for motion with constant acceleration: v = v0 + at; x = x0 + v0 t + (1/2)a t^2; v^2 = v0^2 + 2a(x - x0). |
| position-time graph | A graph with time on the horizontal axis and position on the vertical axis; the slope at any point equals instantaneous velocity. |
| area under the curve | The region bounded by a function and the horizontal axis; on a velocity-time graph it equals displacement, and on an acceleration-time graph it equals change in velocity. |
| component analysis | Decomposing two- or three-dimensional motion into independent one-dimensional kinematic relationships along each axis, solved separately using the same time variable. |
Distance is the total path length (scalar); displacement is the straight-line change in position (vector). An object that returns to its starting point has zero displacement but nonzero distance traveled.
The three kinematic equations only apply when acceleration is constant. If acceleration is a function of time, you must integrate to find velocity and position.
An object moving in a circle at constant speed is still accelerating because the direction of velocity is changing. Acceleration is a vector, so any change in velocity direction counts.
Slope gives the derivative (velocity from a position-time graph; acceleration from a velocity-time graph). Area under the curve gives the integral (displacement from a velocity-time graph). These are not interchangeable.
Relative velocity requires vector addition, not scalar addition. In two-dimensional problems, draw the velocity vectors and add components in each direction separately.
Free-response questions in AP Physics C: Mechanics frequently ask you to move between a graph, an equation, and a written description of the same motion. For Unit 1, that means reading the slope or area of a motion graph, writing the corresponding derivative or integral expression, and explaining what the result means physically.
Expect to differentiate or integrate position, velocity, or acceleration functions given as polynomials or other expressions. You may be asked to find when velocity is zero, when acceleration is maximum, or to determine displacement over a time interval by evaluating a definite integral.
Projectile and relative motion problems require you to decompose vectors into components, write separate equations for each direction, and use time as the link between them. Justify your component setup explicitly and show the kinematic equations you apply in each direction.
Open the individual guides for Unit 1 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsUse unit cheatsheets for a quick visual review after you work through the notes.
open cheatsheetsEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Physics C: Mechanics Unit 1 covers 5 topics in kinematics: Scalars and Vectors, Displacement/Velocity/Acceleration, Representing Motion, Reference Frames and Relative Motion, and Motion in Two or Three Dimensions. Together they build the foundation for analyzing how objects move using mathematical and graphical representations. See the full topic breakdown at AP Physics C: Mechanics Unit 1.
Unit 1: Kinematics makes up 10-15% of the AP Physics C: Mechanics exam. That weight covers motion concepts including scalars and vectors, displacement, velocity, acceleration, reference frames, and two- and three-dimensional motion. It's a smaller unit by percentage, but the skills it builds, especially vector analysis and kinematic equations, show up throughout the rest of the course.
The AP Physics C: Mechanics Unit 1 progress check in AP Classroom includes both MCQ and FRQ parts drawn from all five kinematics topics: Scalars and Vectors, Displacement/Velocity/Acceleration, Representing Motion, Reference Frames and Relative Motion, and Motion in Two or Three Dimensions. The MCQ section tests conceptual understanding and calculation, while the FRQ section asks you to set up and solve multi-part motion problems, often involving graphs or vector components. Practice questions matched to these progress check topics are at AP Physics C: Mechanics Unit 1.
Unit 1 FRQs in AP Physics C: Mechanics focus on kinematics scenarios, typically asking you to derive expressions for displacement, velocity, or acceleration, interpret motion graphs, or analyze two-dimensional projectile motion using vector components. To practice, work through problems that require you to show calculus-based reasoning, write out full solutions with units, and justify each step. Topics like Representing Motion and Motion in Two or Three Dimensions generate the most FRQ-style problems. Find practice FRQs for this unit at AP Physics C: Mechanics Unit 1.
For AP Physics C: Mechanics Unit 1 practice questions, including multiple-choice and practice test problems on kinematics, start at AP Physics C: Mechanics Unit 1. That page has MCQ-style questions covering Scalars and Vectors, Displacement/Velocity/Acceleration, Representing Motion, Reference Frames, and Motion in Two or Three Dimensions, so you can drill each topic or run a full unit practice test.
Start Unit 1 by getting comfortable with vector notation, since scalars and vectors underpin every other topic in kinematics. Then work through displacement, velocity, and acceleration using both calculus definitions and graphs, because AP Physics C: Mechanics expects you to differentiate and integrate position functions, not just use algebra. From there, practice drawing and interpreting motion diagrams for Representing Motion, then move into Reference Frames and two- and three-dimensional problems. A solid study plan looks like this: - Review vector addition and components before anything else. - Derive kinematic relationships using derivatives and integrals, not just memorized formulas. - Sketch position, velocity, and acceleration graphs for the same motion and check they're consistent. - Solve at least five two-dimensional projectile problems with full vector notation. - Time yourself on a short MCQ set to catch gaps before the progress check. All the practice you need for these steps is at AP Physics C: Mechanics Unit 1.