Motion in Two or Three Dimensions
Two- and three-dimensional motion is described by vector-valued position, velocity, and acceleration. Solve components with one shared time, then recombine the components to recover speed, direction, and trajectory.
Topic sequence · 5 of 5
Learning objectives
Objective 1
Represent r⃗(t), v⃗(t), and a⃗(t) with component functions and move among them by differentiation and integration.
Objective 2
Solve two-dimensional projectile motion by separating perpendicular components while enforcing one common time variable.
Objective 3
Determine instantaneous speed and direction from the velocity vector rather than from the position vector or path shape.
Objective 4
Interpret general curved motion using parametric equations, trajectory sketches, component graphs, and limiting-case checks.
- Topic 1.1 vector components and unit vectors.
- Topic 1.2 derivatives/integrals of motion functions.
- Topic 1.3 graph interpretation and constant-acceleration equations.
- Topic 1.4 relative velocity when the observer or medium is moving.
Definitions first
Representations you must read and create
Representation 1
Parametric component equations x(t), y(t), and when useful z(t), all using the same time variable.
Representation 2
Trajectory sketches with velocity tangent to the path and acceleration drawn independently from the tangent.
Representation 3
Separate x(t), y(t), v_x(t), v_y(t), a_x(t), and a_y(t) graphs for projectile motion.
Representation 4
A component table that makes initial values, constant/variable acceleration, and shared time explicit before algebra begins.
Core relationships and calculus
Variables: position components in m
Use: vector position
Variables: velocity components in m/s
Use: componentwise derivative
Variables: acceleration components in m/s²
Use: componentwise derivative
Variables: initial velocity and acceleration function
Use: general vector integration
Variables: initial position and velocity function
Use: general vector displacement
Variables: constant g, negligible drag
Use: ideal projectile component model
Differentiate or integrate vector components independently, but never assign separate times to coordinates of the same object. For projectiles, choose the vertical coordinate equation to determine a physically allowed time, then use that time in the horizontal equation. For general parametric motion, speed is √(v_x²+v_y²+v_z²), and the velocity direction comes from the component ratios.
Write r⃗(t)=x(t)î+y(t)ĵ+z(t)k̂. Differentiation gives v⃗ and then a⃗ componentwise. For ideal projectile motion near Earth, a_x=0 and a_y=−g. Integrating with initial values yields v_x=v₀x, v_y=v₀y−gt, x=x₀+v₀x t, and y=y₀+v₀y t−½gt². Eliminating t between x(t) and y(t) produces a parabolic trajectory only within the physical time interval of the motion.
Use two-dimensional quantitative motion as the main Mechanics model. Treat air resistance only when a force law is supplied; the standard projectile equations assume constant gravitational acceleration and negligible drag.
Physical examples
Example 1
A horizontal projectile keeps constant horizontal velocity while its vertical velocity changes linearly under gravity.
Example 2
At the highest point of a projectile, v_y=0 but v_x may be nonzero, so the total speed usually is not zero.
Example 3
A curved trajectory does not mean acceleration is tangent to the curve; acceleration points in the direction of the velocity change.
Example 4
For a vector function r⃗(t), speed is |dr⃗/dt|, not the derivative of the distance from the origin unless the geometry happens to make those equivalent.
Worked reasoning
Exam reasoning patterns
Reasoning pattern 1
Separate the motion into coordinates before using scalar kinematics; do not insert the total launch speed into both component equations.
Reasoning pattern 2
Use one shared time for every component of the same motion and reject algebraic roots that occur outside the physical interval.
Reasoning pattern 3
Use v⃗, not r⃗, to determine instantaneous direction of travel; velocity is tangent to the trajectory.
Reasoning pattern 4
At a projectile apex, set only the vertical velocity component to zero unless the problem gives a reason for v_x to vanish.
Reasoning pattern 5
When acceleration varies with time or position, return to vector derivatives/integrals instead of forcing constant-acceleration formulas.
Deeper calculus and limiting-case check
For any smooth parametric path, v⃗ is tangent to the trajectory because it is dr⃗/dt. Acceleration can change the speed, the direction, or both. Circular motion in Topic 2.10 is a special case in which a normal/radial component changes direction even when speed is constant.
Component-first workflow for multidimensional motion
Choose axes
Pick axes that simplify the acceleration or geometry. Define the origin and signs before resolving vectors.
Resolve initial vectors
Convert launch speed, velocity, or acceleration into components. Do not reuse the vector magnitude as each component.
Write component models
Use calculus or constant-acceleration relations independently in x, y, and when relevant z.
Share one clock
The coordinates describe one object, so every component uses the same time. Determine t from the component with the useful boundary condition.
Recombine and interpret
Build speed and direction from velocity components, and compare the trajectory/limit with the physical situation.
Vector-calculus map
| Given | Operation | Find | Physical check |
|---|---|---|---|
| r(t) | Differentiate | v(t) | Velocity must be tangent to the trajectory |
| v(t) | Differentiate | a(t) | Acceleration need not point along velocity |
| a(t) | Integrate + initial v | v(t) | Units change from m/s² to m/s |
| v(t) | Integrate + initial r | r(t) | All components use the same interval |
| v components | Magnitude / atan2 | speed / direction | Quadrant and sign must match the sketch |
Interactive projectile analyzer
Ideal projectile: enter launch speed, angle, initial height, and g. The tool finds the positive ground-impact time, horizontal range, maximum height, and impact velocity.
Trajectory and endpoint reasoning
Apex
For an ideal projectile, v_y=0 at the top while a_y=−g. Unless v_x=0 for a special vertical launch, total velocity is not zero.
Impact time
Solve the vertical position equation because the impact height is a boundary condition. Choose the root that lies in the physical post-launch interval.
Range
Once the physical time is known, use the horizontal component model. The horizontal and vertical equations are linked by time, not by identical acceleration.
General curved motion
For r(t), find v(t) and a(t) by differentiation. A curved path can result from acceleration normal to velocity, tangential to velocity, or both.
FRQ + experimental-response architecture
Projectile representation matrix
| Quantity | Horizontal ideal model | Vertical ideal model |
|---|---|---|
| Acceleration | a_x=0 | a_y=−g |
| Velocity | v_x=constant | v_y=v_0y−gt |
| Position | x=x_0+v_0x t | y=y_0+v_0y t−½gt² |
| Graph signature | x(t) linear; v_x(t) horizontal | y(t) quadratic; v_y(t) linear |
| Shared link | The same t describes both coordinates of the same projectile. | |
General parametric synthesis
Suppose r(t)=A cos(ωt) î + B sin(ωt) ĵ. Differentiate to obtain v(t)=−Aω sin(ωt) î + Bω cos(ωt) ĵ and then a(t)=−Aω² cos(ωt) î − Bω² sin(ωt) ĵ. The trajectory is an ellipse when A and B differ. This example separates a broad AP Physics C skill from the projectile special case: component calculus works for any supplied vector function, and the path shape alone does not determine the velocity or acceleration direction without differentiation.
Experimental connection
Video-track a projectile with a meter-scale reference in the plane of motion. Calibrate pixels to meters, identify frame times, fit x(t) and y(t) separately, and inspect residuals. The ideal model predicts an x(t) linear fit and a y(t) quadratic coefficient near −g/2. Discuss frame rate, perspective/parallax, air drag, launch-height uncertainty, and whether the fitted acceleration is consistent with the model.
Connections across Physics C
Connection 1
Topic 1.1 supplies vector decomposition and unit-vector notation.
Connection 2
Topic 1.4 adds moving observers and media to two-dimensional velocity composition.
Connection 3
Topic 2.10 specializes curved motion into radial and tangential acceleration components.
Connection 4
Later force problems reuse the same component discipline: force components determine acceleration components, which then determine the vector motion.
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
Using the total launch speed as both v₀x and v₀y instead of resolving components.
Avoid this
Assigning different time intervals to x and y motion of the same projectile.
Avoid this
Assuming velocity is zero at the highest point when only v_y is zero.
Avoid this
Treating the trajectory direction as the acceleration direction.
Avoid this
Keeping a negative or pre-launch quadratic root without checking the physical time domain.
Avoid this
Applying constant-acceleration equations when the component acceleration 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
Are x and y motions independent?
Their differential equations can be solved separately when the acceleration model separates, but the components share one time and together describe one physical motion.
Why is projectile motion parabolic?
With constant horizontal velocity and constant vertical acceleration, eliminating time from x(t) and y(t) produces a quadratic relation y(x).
Is the velocity zero at the top of a projectile path?
Usually no. The vertical component is zero there, but the horizontal component remains nonzero in the ideal no-drag model.
How do I find instantaneous direction?
Use the velocity components; in two dimensions tanθ_v=v_y/v_x with quadrant handled correctly.
Does Mechanics require full quantitative 3D kinematics?
The primary quantitative emphasis is two-dimensional motion; three-dimensional vector notation is a useful extension and supports later Physics C reasoning.