Unit 1 · Topic 03

One-dimensional motion

Kinematics describes how motion changes without yet explaining the interaction that causes it. Every description requires a reference frame and a sign convention.

Unit 11D motionOpen navigation

Essential The minimum you should retain

Position x locates an object relative to a chosen origin and axis. Displacement is $\Delta x=x_f-x_i$ and depends only on the initial and final positions.

Distance travelled measures total path length and cannot be negative. Distance and displacement magnitude coincide only in particular cases.

UnderstandInterpret and connect

A negative position only means that the object is on the negative side of the origin. It does not indicate its direction of motion.

Changing the origin changes position coordinates but not displacement between two events if axis orientation and scale remain fixed.

DeepenFormulation and conditions

A reference frame includes origin, orientation, spatial scale, and clock. Statements about motion must be understood relative to that frame.

ExploreConnections for further study

Two observers may assign different positions to the same object when they choose different origins. Comparing their descriptions requires transforming coordinates and stating which events are being compared.

Essential The minimum you should retain

Average velocity is displacement divided by the time interval. Its sign gives the direction of the net displacement relative to the axis.

Instantaneous velocity is the rate of change of position. Speed is the magnitude of velocity and is never negative.

UnderstandInterpret and connect

On an $x(t)$ graph, instantaneous velocity is the tangent slope. A large position does not imply a large velocity.

For an out-and-back trip, distance may be large while displacement—and therefore average velocity—is zero.

DeepenFormulation and conditions

The derivative $dx/dt$ arises as the limit of $\Delta x/\Delta t$ as the time interval approaches zero. The position function must be differentiable at the instant considered.

ExploreConnections for further study

If $x(t)$ has a corner, instantaneous velocity may not exist at that exact point even though average velocities exist on both sides. The model must decide whether such an abrupt change is physically reasonable.

v_{med}=\frac{\Delta x}{\Delta t}

Represents

Displacement per unit time over an interval.

Physical interpretation

Its sign represents the direction of the net displacement.

DeepenVariables, conditions, and checks

Variables

$\Delta x$: $x_{f}$ − $x_{i}$ ; usual unit: m
$\Delta t$: $t_{f}$ − $t_{i}$ ; usual unit: s

Conditions of application

$\Delta t$ is positive and both events are described in the same reference frame.

Dimensional check

$L/T$ ; in the SI, $\mathrm{m/s}$ .

Errors it helps prevent

Using distance travelled instead of displacement.

v=\frac{dx}{dt}

Represents

The instantaneous rate of change of position.

Physical interpretation

It is the slope of the tangent to the $x(t)$ graph.

DeepenVariables, conditions, and checks

Variables

v: instantaneous velocity; usual unit: $\mathrm{m/s}$
x: position; usual unit: m
t: time; usual unit: s

Conditions of application

$x(t)$ is differentiable at the instant considered.

Dimensional check

$L/T$ .

Errors it helps prevent

Confusing the value of x with the slope of $x(t)$ .

Essential The minimum you should retain

Acceleration indicates how velocity changes with time. It may change its magnitude, its direction, or both.

Negative acceleration points toward $-x$ ; it does not automatically mean that the object is slowing down.

UnderstandInterpret and connect

Speed increases when velocity and acceleration have the same sign, and decreases when they have opposite signs.

On a $v(t)$ graph, slope is acceleration. On an $a(t)$ graph, the function value is acceleration, not its slope.

DeepenFormulation and conditions

Instantaneous acceleration is $dv/dt$ and is also the second derivative of position when $x(t)$ is twice differentiable.

ExploreConnections for further study

In more than one dimension, acceleration can change the direction of velocity without changing its magnitude. The one-dimensional sign rule must then be replaced by a vector comparison.

a_{med}=\frac{\Delta v}{\Delta t}

Represents

The change in velocity per unit time over an interval.

Physical interpretation

Its sign indicates how the velocity component changes along the chosen axis.

DeepenVariables, conditions, and checks

Variables

$\Delta v$: $v_{f}$ − $v_{i}$ ; usual unit: $\mathrm{m/s}$

Conditions of application

The velocities are expressed relative to the same axis and frame.

Dimensional check

$L/T^2$ ; in the SI, $\mathrm{m/s^2}$ .

Errors it helps prevent

Interpreting negative acceleration as an automatic decrease in speed.

a=\frac{dv}{dt}=\frac{d^2x}{dt^2}

Represents

The instantaneous rate of change of velocity.

Physical interpretation

It is the slope of $v(t)$ .

DeepenVariables, conditions, and checks

Variables

a: instantaneous acceleration; usual unit: $\mathrm{m/s^2}$
v: velocity; usual unit: $\mathrm{m/s}$
x: position; usual unit: m
t: time; usual unit: s

Conditions of application

$v(t)$ is differentiable; for the second equality, $x(t)$ is twice differentiable.

Dimensional check

$L/T^2$ .

Essential The minimum you should retain

The slope of $x(t)$ is $v(t)$ , and the slope of $v(t)$ is $a(t)$ . These relations connect the shapes of the three graphs.

Signed area under $v(t)$ represents displacement; signed area under $a(t)$ represents change in velocity. Regions below the axis contribute negatively.

UnderstandInterpret and connect

Differentiation follows local changes: $x(t)$ → $v(t)$ → $a(t)$ . Integration accumulates changes: $a(t)$ → $v(t)$ → $x(t)$ , together with initial conditions.

A horizontal line on $x(t)$ indicates rest; a horizontal line on $v(t)$ indicates constant velocity, which need not be zero.

DeepenFormulation and conditions

The areas are definite integrals. Their physical interpretation includes sign and unit: ( $\mathrm{m/s}$ )·s gives m, while ( $\mathrm{m/s^2}$ )·s gives $\mathrm{m/s}$ .

ExploreConnections for further study

Experimental graphs contain noise and do not always provide exact slopes or areas. In those cases, trends are estimated and the result's uncertainty is communicated.

$x(t)$ — its slope is $v(t)$

x(t)

The position curve becomes progressively steeper: velocity is positive and increases with time.

$v(t)$ — slope: a; signed area: $\Delta x$

v(t)

Velocity increases linearly. Its constant slope is 1 $\mathrm{m/s^2}$ , and the accumulated signed area equals the displacement.

$a(t)$ — its signed area is $\Delta v$

a(t)

Acceleration is constant and positive. The rectangle's area between two instants represents the increase in velocity over that interval.

Concept review

Common errors

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

Confusing position with displacement.

Position locates an event; displacement compares final and initial positions.

Confusing distance with displacement.

Distance adds the path travelled; displacement depends only on the endpoints and has a sign.

Confusing speed with velocity.

Speed is the nonnegative magnitude of velocity; velocity includes direction and sense.

Interpreting x < 0 as motion toward

-x

x < 0 only locates the object. The sign of v determines the direction of motion.

Interpreting v < 0 as slowing down.

The sign of v indicates direction. Compare the signs of v and a to determine whether speed decreases.

Interpreting a < 0 as slowing down.

a < 0 indicates a direction toward $-x$ . If v is also negative, speed increases.

Confusing slope with the value of a function on a graph.

Distinguish the point's height from its local inclination: on $x(t)$ , the slope—not x—is v.

Using distance to calculate average velocity.

Average velocity uses displacement. Distance/time is average speed.

Ignoring the reference frame.

State the origin, axis orientation, and clock before assigning signs or comparing motions.

Aula Física

One-dimensional motion

Reference frame, position, and displacement

Essential The minimum you should retain

Average velocity, instantaneous velocity, and speed

Essential The minimum you should retain

Average velocity in one dimension

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Instantaneous velocity as the derivative of position

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Average and instantaneous acceleration; signs

Essential The minimum you should retain

Average acceleration in one dimension

Variables

Conditions of application

Dimensional check

Errors it helps prevent

Instantaneous acceleration as the derivative of velocity and second derivative of position

Variables

Conditions of application

Dimensional check

Relationships among x(t), v(t), and a(t)

Essential The minimum you should retain

Common errors