Unit 1 · Topic 05

Motion in two and three dimensions

In two and three dimensions, position, velocity, and acceleration are vectors. Cartesian components are related through the same time.

Unit 12D/3D motionOpen navigation

Concept 01

Position, velocity, and acceleration vectors

Essential The minimum you should retain

The position vector r→(t) \vec r(t) locates the particle. Velocity v→=dr→dt \vec v=d\vec r/dt is tangent to the trajectory, and acceleration a→=dv→dt \vec a=d\vec v/dt describes the change in the velocity vector.

Each component may be differentiated separately in a fixed Cartesian basis.

UnderstandInterpret and connect

A curved trajectory may have constant speed and still have acceleration because the direction of velocity changes.

The x, y, and z components do not represent motions with different times: they describe the same event and share t.

DeepenFormulation and conditions

Componentwise differentiation assumes that Cartesian unit vectors are constant. In moving bases such as polar coordinates, the unit vectors must also be differentiated.

ExploreConnections for further study

Trajectory curvature depends on how the direction of velocity changes. Two motions may follow the same curve at different rates and therefore have different accelerations.

Mathematical relation

Cartesian position vector in three dimensions

r→(t)=x(t)i^+y(t)j^+z(t)k^ \vec r(t)=x(t)\hat i+y(t)\hat j+z(t)\hat k
Represents

The particle's location relative to the Cartesian origin.

Physical interpretation

The coordinates are the projections of r onto the chosen Cartesian basis.

DeepenVariables, conditions, and checks

Variables

r
position vector; usual unit: m
x, y, z
Cartesian coordinates; usual unit: m
t
time; usual unit: s

Conditions of application

  • i, j, and k form a fixed Cartesian basis.

Dimensional check

Each component has the dimension of length.

Mathematical relation

Velocity vector as the derivative of position

v→=dr→dt \vec v=\frac{d\vec r}{dt}
Represents

The rate of change of the position vector.

Physical interpretation

v is tangent to the trajectory when the speed is nonzero.

DeepenVariables, conditions, and checks

Variables

v
velocity vector; usual unit: ms \mathrm{m/s}
r
position vector; usual unit: m
t
time; usual unit: s

Conditions of application

  • r→(t) \vec r(t) is differentiable at the instant considered.

Dimensional check

LT L/T .

Mathematical relation

Acceleration vector as the derivative of velocity

a→=dv→dt \vec a=\frac{d\vec v}{dt}
Represents

The rate of change of the velocity vector.

Physical interpretation

It can change speed, direction, or both.

DeepenVariables, conditions, and checks

Variables

a
acceleration vector; usual unit: ms2 \mathrm{m/s^2}
v
velocity vector; usual unit: ms \mathrm{m/s}
t
time; usual unit: s

Conditions of application

  • v(t) v(t) is differentiable at the instant considered.

Dimensional check

LT2 L/T^2 .

Concept 02

Parallel and perpendicular components

Essential The minimum you should retain

The parallel component of acceleration changes speed. The perpendicular component changes the direction of velocity.

Purely perpendicular acceleration can curve a trajectory without changing instantaneous speed.

UnderstandInterpret and connect

This decomposition follows the trajectory and need not coincide with the x and y axes. It is especially useful in circular motion.

DeepenFormulation and conditions

For nonzero speed, tangential acceleration has magnitude dvdt dv/dt and normal acceleration has magnitude v2 v^2 /ρ, where ρ is the local radius of curvature.

ExploreConnections for further study

The same tangential-normal decomposition applies to any smooth trajectory. The local radius of curvature replaces the fixed radius of a circle.

Concept 03

Projectile motion

Essential The minimum you should retain

In the model without air resistance and with approximately constant g, ax a_{x} = 0 and ay a_{y} = −g when +y points upward.

Horizontal motion has constant velocity and vertical motion has constant acceleration; both share the same time.

UnderstandInterpret and connect

At the highest point, the vertical velocity component is zero, but horizontal velocity may be nonzero and acceleration still points downward.

The parabolic shape results from combining x linear in t with y quadratic in t under these conditions.

DeepenFormulation and conditions

Range or flight-time formulas that assume equal initial and final heights must not be applied to launches between different heights.

ExploreConnections for further study

Air resistance couples horizontal and vertical motion because it depends on the velocity vector. The trajectory is no longer exactly parabolic, and range requires a more complete model.

Mathematical relation

Acceleration components of an ideal projectile

ax=0,ay=−g a_x=0,\quad a_y=-g
Represents

Acceleration in the ideal projectile model.

Physical interpretation

Horizontal acceleration is zero; vertical acceleration points downward throughout the flight.

DeepenVariables, conditions, and checks

Variables

ax a_{x} , ay a_{y}
acceleration components; usual unit: ms2 \mathrm{m/s^2}
g
local gravitational acceleration magnitude; usual unit: ms2 \mathrm{m/s^2}

Conditions of application

  • Negligible air resistance, approximately constant g, and +y directed upward.

Dimensional check

ax a_{x} , ay a_{y} , and g have dimension LT2 L/T^2 .

Errors it helps prevent

  • Assuming that ay a_{y} becomes zero at the highest point.

Mathematical relation

Component form of an ideal projectile's position

x=x0+v0xt,y=y0+v0yt−12gt2 x=x_0+v_{0x}t,\quad y=y_0+v_{0y}t-\frac12gt^2
Represents

Horizontal and vertical position as functions of the same time.

Physical interpretation

x(t) x(t) is linear and y(t) is quadratic; eliminating t gives a parabola.

DeepenVariables, conditions, and checks

Variables

x, y, x0 x_{0} , y₀
final and initial positions; usual unit: m
v0 v_{0} ₓ, v0 v_{0} ᵧ
initial velocity components; usual unit: ms \mathrm{m/s}
t
elapsed time; usual unit: s
g
local gravitational acceleration magnitude; usual unit: ms2 \mathrm{m/s^2}

Conditions of application

  • The same ideal-projectile assumptions and fixed Cartesian axes.

Dimensional check

Every position term has dimension L.

Errors it helps prevent

  • Using different times for the horizontal and vertical components.
A parabolic trajectory starts at the ground and returns to it, with points advancing at equal time intervals.groundvᵧ = 0; aᵧ = −g
  • Trajectory

Each point corresponds to the same time in x and y. Uniform horizontal spacing reflects constant vx v_{x} ; vertical curvature reflects ay a_{y} = −g.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Treating x and y as motions with independent times.

The components are analysed separately, but they describe the same event and share t.

Assigning horizontal acceleration in the ideal model.

Without air resistance, ax a_{x} = 0. Acceleration is vertical and points downward.

Believing that a = 0 at the highest point.

The vertical velocity component becomes zero, not gravitational acceleration.

Confusing zero vertical velocity with zero total velocity.

At the highest point vy v_{y} = 0, but vx v_{x} may still be nonzero.