2 - One-Dimensional Kinematics

A progressive guide to describing one-dimensional motion using position, displacement, velocity, acceleration, graphs, constant-acceleration equations, and free-fall models.

, , and distance

Motion in one dimension occurs along a single straight line. Before using equations or graphs, establish a coordinate system:

Choose an origin, where the coordinate is zero.
Choose a positive direction.
Choose a unit, usually meters.

The coordinate describes where the object is, not how far it has traveled. For example, if east is positive, $x=+8\,\text{m}$ means $8\,\text{m}$ east of the origin, while $x=-3\,\text{m}$ means $3\,\text{m}$ west of the origin.

compares the final and initial positions:

\Delta x=x_f-x_i

Because is signed, it includes direction. Distance is the total length of the path and cannot be negative. If a student walks from $x=2\,\text{m}$ to $x=10\,\text{m}$ , then returns to $x=6\,\text{m}$ , the is $+4\,\text{m}$ , whereas the distance is $8\,\text{m}+4\,\text{m}=12\,\text{m}$ .

Takeaway: identifies location; compares locations; distance counts the entire path.

Velocity and

describes the net change in per elapsed time:

\bar v=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}

Velocity is signed. A positive value indicates motion in the positive coordinate direction, and a negative value indicates motion in the negative direction. describes motion at one particular instant:

v=\frac{dx}{dt}

On a -versus-time graph, is the slope of the tangent line. For a straight section, the slope can be calculated as $v=\frac{\Delta x}{\Delta t}$ .

is the magnitude of velocity:

\text{speed}=|v|

has no direction and is never negative. An object that returns to its starting point can have zero but a nonzero average .

Takeaway: Use velocity when direction matters and when only the magnitude of motion matters.

and changes in motion

measures how velocity changes with time. Average is

\bar a=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t_f-t_i}

Instantaneous is

a=\frac{dv}{dt}

The SI unit is $\text{m/s}^2$ . An of $+2.0\,\text{m/s}^2$ means that velocity becomes $2.0\,\text{m/s}$ more positive each second.

The signs of velocity and must be considered together:

If $v>0$ and $a<0$ , the object moves in the positive direction and slows down.
If $v<0$ and $a>0$ , the object moves in the negative direction and slows down.
If $v>0$ and $a>0$ , the object moves in the positive direction and speeds up.
If $v<0$ and $a<0$ , the object moves in the negative direction and speeds up.

Thus, negative does not automatically mean slowing down. 's sign indicates direction relative to the coordinate system.

Takeaway: An object slows down when velocity and have opposite signs; it speeds up when they have the same sign.

Reading motion graphs

Graphs connect a motion quantity to its rate of change or accumulated effect.

For a -versus-time graph:

The vertical coordinate gives .
The slope gives velocity.
A positive slope means positive velocity.
A negative slope means negative velocity.
A horizontal line means the object is at rest.
A changing slope means velocity is changing, so the object is accelerating.

For a velocity-versus-time graph:

The vertical coordinate gives velocity.
The slope gives .
A horizontal line represents constant velocity and zero .
The signed area gives :

\Delta x=\text{area under a }v\text{-versus-}t\text{ graph}

Areas above the time axis contribute positively, and areas below it contribute negatively. To find total distance, add the magnitudes of the separate areas so that opposite-direction motion does not cancel.

For an -versus-time graph, the vertical coordinate gives , and the area gives the change in velocity:

\Delta v=\text{area under an }a\text{-versus-}t\text{ graph}

A appears as a horizontal line on this graph.

Takeaway: Slopes describe rates of change, while areas describe accumulated changes.

Constant- equations

When is present, use the equation that contains the known quantities and the desired unknown. The main equations are

v_f=v_i+at

x_f=x_i+v_i t+\frac{1}{2}at^2

v_f^2=v_i^2+2a(x_f-x_i)

An equivalent equation is

\Delta x=v_i t+\frac{1}{2}at^2

and a useful average-velocity form is

\Delta x=\left(\frac{v_i+v_f}{2}\right)t

The variables are initial $x_i$ , final $x_f$ , $\Delta x=x_f-x_i$ , initial velocity $v_i$ , final velocity $v_f$ , $a$ , and elapsed time $t$ .

Choose efficiently:

If time is known and final velocity is requested, use $v_f=v_i+at$ .
If time is known and final is requested, use $x_f=x_i+v_i t+\frac{1}{2}at^2$ .
If time is not given, use $v_f^2=v_i^2+2a\Delta x$ .

For example, a car with $v_i=4.0\,\text{m/s}$ and $a=3.0\,\text{m/s}^2$ for $t=5.0\,\text{s}$ reaches

v_f=(4.0\,\text{m/s})+(3.0\,\text{m/s}^2)(5.0\,\text{s})=19.0\,\text{m/s}

Its is

\Delta x=(4.0)(5.0)+\frac{1}{2}(3.0)(5.0)^2=57.5\,\text{m}

Takeaway: State the sign convention first, then select an equation that avoids unnecessary unknowns.

and vertical motion

is modeled as constant- motion when gravity is the only significant force and air resistance is negligible. Near Earth's surface, the downward has magnitude

g\approx 9.8\,\text{m/s}^2

For many AP Physics calculations, $g$ is approximated as $10\,\text{m/s}^2$ . If upward is positive, then

a_y=-g

and the equations become

v_{yf}=v_{yi}-gt

y_f=y_i+v_{yi}t-\frac{1}{2}gt^2

v_{yf}^2=v_{yi}^2-2g(y_f-y_i)

An object thrown upward has positive initial velocity under this convention, negative throughout its flight, and zero at its highest point. Its is still $-g$ at that point. During its downward motion, velocity is negative and increases.

For a rock dropped from rest at $y_i=20\,\text{m}$ , with the ground at $y_f=0$ and upward positive,

0=20-\frac{1}{2}(9.8)t^2

t=\sqrt{\frac{40}{9.8}}\approx 2.0\,\text{s}

The final velocity is approximately $-19.6\,\text{m/s}$ , where the negative sign indicates downward motion.

Takeaway: At the highest point of an upward throw, velocity is zero but gravitational is not zero.

A reliable solution strategy

Use this sequence for a one-dimensional kinematics problem:

Draw a diagram showing the initial and final states.
Choose the origin and positive direction.
List known quantities and the unknown, including signs and units.
Decide whether is constant.
Select an equation containing the desired unknown.
Substitute values with units.
Check the result's units, sign, and physical reasonableness.

Do not substitute when an equation requires velocity. has no sign, whereas velocity must be assigned a sign using the chosen coordinate system. A negative result is not automatically an error; it may indicate motion or in the negative direction.

The central relationships are:

-versus-time slope gives velocity.
Velocity-versus-time slope gives .
Velocity-versus-time area gives .
-versus-time area gives change in velocity.
Constant- equations connect , velocity, , and time only when is constant.

Final takeaway: A consistent coordinate system turns direction into signs, allowing words, graphs, and equations to describe the same one-dimensional motion.

Position, displacement, and distance

Velocity and speed

Acceleration and changes in motion