Representing Motion: Position, Velocity, and Acceleration in One Dimension

Representing Motion

Coordinate Systems and Position

To analyze motion, we use coordinate axes to define positions in space. The x-axis is typically used for horizontal motion (positive to the right), and the y-axis for vertical motion (positive upward). The choice of axes is flexible, but must be clearly indicated in your work.

• Position is the location of an object relative to an origin (0 point) on the axis.

• Positions to the right/above the origin are positive; to the left/below are negative.

Motion Diagrams

A motion diagram visually represents an object's position at successive time intervals. Each dot corresponds to the object's position at a particular time, and arrows indicate direction and relative speed.

Position vs. Time Graphs

Plotting position (x) versus time (t) provides a graphical representation of motion. The slope of the graph at any point indicates the object's velocity.

• A straight line indicates constant velocity.

• A curve indicates changing velocity (acceleration).

From Position to Velocity

Understanding Slope and Velocity

The slope of a position-versus-time graph gives the velocity of the object. A steeper slope means a higher speed.

• Average velocity is the total displacement divided by the total time interval.

• Mathematically: $v_{av} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

• Velocity is a vector (has both magnitude and direction).

Examples and Applications

Consider a skateboard's position tracked over time. The velocity at a specific time can be found by calculating the slope of the tangent to the curve at that point.

• If the graph is flat (slope = 0), the object is at rest.

• Positive slope: moving in the positive direction; negative slope: moving in the negative direction.

Summary Table: Reading Position vs Time Graphs

Graph Feature	Physical Meaning
Above x-axis	Positive position
Below x-axis	Negative position
Straight line	Constant velocity
Curved line	Changing velocity (acceleration)

Velocity vs. Time Graphs

Constructing Velocity Graphs

A velocity-versus-time graph can be deduced from the position-versus-time graph. The value of velocity at each instant is the slope of the position graph at that time.

• Constant velocity: horizontal line.

• Changing velocity: sloped or curved line.

Displacement from Velocity Graphs

The displacement ($\Delta x$) over a time interval is equal to the area under the velocity-versus-time graph during that interval.

• For constant velocity, area is a rectangle: $\Delta x = v \Delta t$

• For changing velocity, area may be more complex (e.g., triangle, trapezoid).

Instantaneous Velocity

Definition and Calculation

Instantaneous velocity is the velocity of an object at a specific instant in time. It is found as the slope of the tangent to the position-versus-time curve at that point.

• For a curve, draw a tangent line at the desired point and calculate its slope.

• Mathematically: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

Acceleration

Average and Instantaneous Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity, with SI units of m/s2.

• Average acceleration: $a_{av} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$

• Acceleration can be positive or negative, depending on whether the object is speeding up or slowing down and the direction of motion.

Example Problem: Lion's Acceleration

• A lion accelerates from rest at $9.5\ \mathrm{m/s}^2$ for 1 second. How long to reach $4.5\ \mathrm{m/s}$?

• Known: $a = 9.5\ \mathrm{m/s}^2$, $v_i = 0$, $v_f = 4.5\ \mathrm{m/s}$

• Equation: $v_f = v_i + a t$

• Solve for $t$: $t = \frac{v_f - v_i}{a} = \frac{4.5}{9.5} \approx 0.47\ \mathrm{s}$

Example Problem: Skater's Acceleration

• Alice skates east at $3.20\ \mathrm{m/s}$, then accelerates west at $0.8\ \mathrm{m/s}^2$.

• Find final velocity after 3.0 s and 6.0 s, and describe the motion.

• Equation: $v_f = v_i + a t$

• Direction matters: acceleration is opposite to initial velocity.

Summary Table: Key Kinematic Quantities

Quantity	Definition	SI Unit
Position ($x$)	Location relative to origin	m
Displacement ($\Delta x$)	Change in position	m
Velocity ($v$)	Rate of change of position	m/s
Acceleration ($a$)	Rate of change of velocity	m/s2