Unit 1 · Topic 04

Equations of motion

Kinematic equations express a model. Before using them, state the reference frame, interval, and acceleration behaviour.

Unit 1EquationsOpen navigation

Essential The minimum you should retain

If acceleration is constant over the interval, velocity changes linearly and position changes quadratically with time.

The four usual equations are relations from the same model; choose one after identifying data, unknown, and conditions, not through a superficial match of symbols.

UnderstandInterpret and connect

On $v(t)$ , the slope is a and the signed area is $\Delta x$ . When v crosses zero, the object changes direction if the description continues smoothly.

The average velocity $(v_0+v)/2$ may be used in this form only when acceleration is constant.

DeepenFormulation and conditions

Integrating constant a gives $v=v_0+at$ . Integrating that velocity from the initial condition gives $x=x_0+v_0t+\frac12at^2$ . The other relations follow by eliminating t or using the linear average of velocities.

ExploreConnections for further study

A real motion may be approximated by intervals of nearly constant acceleration. The approximation is useful only when each interval is short enough for the purpose of the analysis.

v=v_0+at

Represents

Velocity after an interval t with constant acceleration.

Physical interpretation

$v(t)$ is a straight line whose slope is a.

DeepenVariables, conditions, and checks

Variables

v, $v_{0}$: final and initial velocities; usual unit: $\mathrm{m/s}$
a: constant acceleration; usual unit: $\mathrm{m/s^2}$
t: elapsed time; usual unit: s

Conditions of application

a is constant throughout the interval and t is measured from the initial condition.

Dimensional check

Every term has dimension $L/T$ .

Errors it helps prevent

Using it when a changes with time.

x=x_0+v_0t+\frac12at^2

Represents

Position after an interval t with constant acceleration.

Physical interpretation

$x(t)$ is quadratic; its local slope equals $v(t)$ .

DeepenVariables, conditions, and checks

Variables

x, $x_{0}$: final and initial positions; usual unit: m
$v_{0}$: initial velocity; usual unit: $\mathrm{m/s}$
a: constant acceleration; usual unit: $\mathrm{m/s^2}$
t: elapsed time; usual unit: s

Conditions of application

a is constant and $x_{0}$ , $v_{0}$ correspond to the start of the interval.

Dimensional check

$x_{0}$ , $v_{0}$ t, and $1/2$ a $t^2$ have the dimension of length.

Errors it helps prevent

Omitting the initial position or losing the sign of a.

v^2=v_0^2+2a(x-x_0)

Represents

The relation between velocity and displacement without using time explicitly.

Physical interpretation

It retains information about $v^2$ ; the direction of v must be determined from context.

DeepenVariables, conditions, and checks

Variables

v, $v_{0}$: final and initial velocities; usual unit: $\mathrm{m/s}$
a: constant acceleration; usual unit: $\mathrm{m/s^2}$
x, $x_{0}$: final and initial positions; usual unit: m

Conditions of application

a is constant over the displacement considered.

Dimensional check

Both sides have dimension L²/T².

Errors it helps prevent

Automatically taking the positive root when solving for v.

\Delta x=\frac{v_0+v}{2}t

Represents

The signed area under a velocity that changes linearly.

Physical interpretation

$(v_0+v)/2$ is the average velocity because $v(t)$ is linear.

DeepenVariables, conditions, and checks

Variables

$\Delta x$: displacement; usual unit: m
$v_{0}$ , v: initial and final velocities; usual unit: $\mathrm{m/s}$
t: elapsed time; usual unit: s

Conditions of application

Only for constant acceleration throughout the interval.

Dimensional check

( $L/T$ )·T = L.

Errors it helps prevent

Using this average when acceleration is not constant.

v(t)=4−1.2t

Velocity $v(t)$ =4−1.2t crosses zero: the positive area before the crossing contributes displacement toward $+x$ , while the later area contributes displacement toward $-x$ .

Essential The minimum you should retain

A change of direction occurs when velocity changes sign. The instant v = 0 separates two intervals of motion, but acceleration may remain nonzero.

Displacement adds signed contributions; distance adds lengths and therefore requires separating intervals when the object reverses.

UnderstandInterpret and connect

Acceleration opposite to velocity reduces speed to instantaneous rest. If acceleration continues, speed then increases in the opposite direction.

DeepenFormulation and conditions

To calculate distance from $v(t)$ , locate its zeros and add the absolute value of displacement over each interval. Integrating | $v(t)$ | gives distance travelled.

ExploreConnections for further study

Distance travelled is the total variation of position over the interval. The idea generalizes to curved paths by adding increasingly small path lengths.

Essential The minimum you should retain

In the introductory free-fall model, air resistance is neglected, g is approximated as constant, and a region near Earth's surface is studied.

g ≈ 9.8 $\mathrm{m/s^2}$ is the approximate local magnitude. If +y points upward, $a_{y}$ = −g. The sign comes from the axis, not from the symbol g.

UnderstandInterpret and connect

At the maximum height of a vertical launch, $v_{y}$ = 0 but $a_{y}$ = −g. Gravity acts during ascent, at the highest point, and during descent.

Time or speed symmetries apply only between points at the same height under the same model conditions.

DeepenFormulation and conditions

The body may be treated as a particle when its size and rotation are irrelevant to the question. If air resistance matters, acceleration is no longer constant or necessarily equal during ascent and descent.

ExploreConnections for further study

With air resistance, objects of the same shape may fall differently depending on mass, area, and speed. The constant value −g is no longer the total acceleration, although gravity still acts.

y(t) during a vertical launch

y(t)

Height increases until the slope becomes zero and then decreases. The downward concavity reflects $a_{y}$ = −g throughout the interval.

$v_{y}$ (t) under gravitational acceleration

vᵧ(t)

Vertical velocity decreases linearly. It crosses zero at maximum height, but its slope remains −g.

Essential The minimum you should retain

When a changes with time, the constant-acceleration equations are not valid. Velocity is obtained by accumulating $a(t)$ , and position by accumulating $v(t)$ .

Initial conditions determine which particular motion corresponds to those functions.

UnderstandInterpret and connect

The integral of $a(t)$ between two instants is $\Delta v$ , not necessarily the final velocity. The initial velocity must be added.

Similarly, the integral of $v(t)$ is displacement and is added to the initial position to obtain $x(t)$ .

DeepenFormulation and conditions

If acceleration depends on position or velocity, changing variables or solving a differential equation may be necessary. This unit establishes the conceptual relation without yet constructing a general solution method.

ExploreConnections for further study

When data are known only at discrete instants, accumulation can be approximated numerically with rectangle or trapezoid areas. Step size controls part of the error.

v(t)=v_0+\int_{t_0}^{t}a(\tau)d\tau

Represents

Velocity as the initial condition plus the accumulated change.

Physical interpretation

The signed area under $a(t)$ is $\Delta v$ , not v by itself.

DeepenVariables, conditions, and checks

Variables

$v(t)$ , $v_{0}$: final and initial velocities; usual unit: $\mathrm{m/s}$
a(τ): integrated acceleration; usual unit: $\mathrm{m/s^2}$
t, t₀, τ: times and integration variable; usual unit: s

Conditions of application

$a(t)$ is integrable over the interval and $v_{0}$ corresponds to t₀.

Dimensional check

( $L/T^2$ )·T = $L/T$ .

x(t)=x_0+\int_{t_0}^{t}v(\tau)d\tau

Represents

Position as the initial condition plus the accumulated displacement.

Physical interpretation

The signed area under $v(t)$ is $\Delta x$ .

DeepenVariables, conditions, and checks

Variables

$x(t)$ , $x_{0}$: final and initial positions; usual unit: m
v(τ): integrated velocity; usual unit: $\mathrm{m/s}$
t, t₀, τ: times and integration variable; usual unit: s

Conditions of application

$v(t)$ is integrable over the interval and $x_{0}$ corresponds to t₀.

Dimensional check

( $L/T$ )·T = L.

Concept review

Common errors

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

Applying kinematic equations when a is not constant.

Check the behaviour of a first. If it changes, use the appropriate differential or integral relation.

Choosing a formula only because it contains the known variables.

Identify the model, intervals, and assumptions before selecting a mathematical relation.

Using

(v_0+v)/2

outside constant acceleration.

That average works because $v(t)$ is linear when a is constant. Otherwise, you must integrate.

Ignoring a change of direction when calculating distance.

Find when v = 0 and add the path length in each interval separately.

Assuming that v = 0 implies a = 0.

Rest can be instantaneous. Acceleration describes how v changes and can remain nonzero.

Treating g as necessarily negative.

g denotes an approximately positive magnitude. The component's sign depends on the chosen axis.

Believing gravity acts only while the object moves downward.

In the model, gravitational acceleration acts during ascent, maximum height, and descent.

Believing that a = 0 at the highest point.

At that instant $v_{y}$ = 0, but if +y points upward, $a_{y}$ = −g.

Applying symmetry between points at different heights without checking the conditions.

Simple symmetry compares points at equal height within the same model and without air resistance.

Aula Física

Equations of motion

Constant-acceleration model

Essential The minimum you should retain

Velocity for constant acceleration

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Position for constant acceleration

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Time-independent kinematic relation for constant acceleration

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Displacement from average velocity with constant acceleration

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Change of direction and interval-by-interval analysis

Essential The minimum you should retain

Free fall as a model

Essential The minimum you should retain

Nonconstant acceleration and integration

Essential The minimum you should retain

Velocity obtained by integrating acceleration

Variables

Conditions of application

Dimensional check

Position obtained by integrating velocity

Variables

Conditions of application

Dimensional check

Common errors