Describing Motion: Position, Velocity, and Acceleration (Ch 2.1–2.4)

Describing Motion

Big Ideas

Understanding motion requires precise definitions and mathematical tools. In physics, vectors are used to describe position, displacement, and velocity. We distinguish between average and instantaneous quantities, and use graphs to visualize relationships between these variables.

• Vectors represent quantities with both magnitude and direction, such as displacement and velocity.

• Scalars have only magnitude, such as distance and speed.

• Motion can be described using position, displacement, velocity, and acceleration.

Position, Distance, and Displacement

Definitions and Examples

Position describes the location of an object relative to a chosen origin. Displacement is the change in position, while distance is the total path length traveled.

• Position (x): The location of an object along a coordinate axis.

• Displacement (\( \Delta x \)): The change in position, a vector quantity.

  • \( \Delta x = x_{final} - x_{initial} \)

• Distance: The total length of the path traveled, a scalar quantity.

Example: If you walk 60 m east, then 80 m west, your total distance is 140 m, but your displacement is -20 m (if east is positive).

Average Speed and Average Velocity

Definitions and Calculations

Average speed and average velocity describe how fast and in what direction an object moves over a time interval.

• Average Speed: Total distance traveled divided by total time.

  • \( \text{Average speed} = \frac{\text{Total distance}}{\Delta t} \)

• Average Velocity: Displacement divided by total time.

  • \( \text{Average velocity} = \frac{\Delta x}{\Delta t} \)

Example: If you travel 140 m in 10 minutes (600 s), your average speed is \( \frac{140\ \text{m}}{600\ \text{s}} = 0.233\ \text{m/s} \). If your displacement is 20 m, your average velocity is \( \frac{20\ \text{m}}{600\ \text{s}} = 0.033\ \text{m/s} \).

Position vs. Time Graphs

Interpreting Graphs

Position-time graphs visually represent how an object's position changes over time. The slope of the graph gives the velocity.

• Total Distance Traveled: Sum of the absolute values of all position changes.

• Displacement: Difference between final and initial positions.

• Velocity: Slope of the position vs. time graph.

  • \( v = \frac{\Delta x}{\Delta t} \)

Example: If an object moves from 0 m to 60 m in 4 s, then back to 20 m in the next 8 s, the total distance is 60 m + 40 m = 100 m, displacement is 20 m, and average velocity is \( \frac{20\ \text{m}}{12\ \text{s}} = 1.67\ \text{m/s} \).

Velocity vs. Time Graphs

Drawing and Interpreting

Velocity-time graphs show how velocity changes with time. The slope of this graph represents acceleration, and the area under the curve gives displacement.

• Constant Velocity: Horizontal line (zero slope).

• Changing Velocity: Sloped line; the slope equals acceleration.

• Displacement: Area under the velocity-time curve.

  • \( \Delta x = \int v\,dt \) (for calculus-based analysis)

Instantaneous Velocity

Definition and Calculation

Instantaneous velocity is the velocity of an object at a specific moment in time. It is the derivative of position with respect to time.

• Instantaneous Velocity (v):

  • \( v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt} \)

• On a position-time graph, it is the slope of the tangent at a point.

Acceleration

Definition and Graphical Interpretation

Acceleration is the rate at which velocity changes with time. On a velocity-time graph, acceleration is the slope of the line.

• Average Acceleration (a):

  • \( a = \frac{\Delta v}{\Delta t} \)

• Instantaneous Acceleration:

  • \( a = \frac{dv}{dt} \)

Constant Acceleration

Kinematic Equations

When acceleration is constant, the following kinematic equations describe motion:

• \( v = v_0 + at \)

• \( x = x_0 + v_0 t + \frac{1}{2} a t^2 \)

• \( v^2 = v_0^2 + 2a(x - x_0) \)

Legend for above:

• \( v_0 \): Initial velocity

• \( v \): Final velocity

• \( x_0 \): Initial position

• \( x \): Final position

• \( a \): Acceleration

• \( t \): Time

Example: Airplane Takeoff

A Boeing 747 accelerates at 2.6 m/s2 to reach a takeoff speed of 70 m/s.

• (a) Time to reach takeoff speed:

  • \( v = v_0 + at \Rightarrow t = \frac{v - v_0}{a} = \frac{70\ \text{m/s} - 0}{2.6\ \text{m/s}^2} = 26.9\ \text{s} \)

• (b) Minimum runway length:

  • \( x = x_0 + v_0 t + \frac{1}{2} a t^2 \Rightarrow x = 0 + 0 + \frac{1}{2} (2.6) (26.9)^2 = 940\ \text{m} \)