Kinematics in One Dimension: Position, Velocity, and Calculus in Motion

Ch. 2: Kinematics in One Dimension

Uniform Motion

Kinematics in one dimension studies the motion of objects along a straight line, focusing on position, velocity, and acceleration. Uniform motion refers to motion at a constant velocity, where the object's displacement is the same during equal time intervals.

• Position (x): The location of an object along a straight line, measured from a reference point.

• Displacement (\( \Delta x \)): The change in position over a time interval \( \Delta t \).

• Uniform Velocity (v): The rate of change of position, given by the slope of the position-time graph.

Key Equation:

$v = \frac{\Delta x}{\Delta t}$

Position and Velocity Graphs

The position-versus-time graph for uniform motion is a straight line, and its slope represents the velocity. The velocity-versus-time graph is a horizontal line for uniform motion, indicating constant velocity.

• Positive Slope: Object moves in the positive direction.

• Negative Slope: Object moves in the negative direction.

• Zero Slope: Object is at rest.

Example: Solving Uniform Motion Problems

To solve problems involving two objects moving toward each other, set up equations for each object's position as a function of time and solve for the meeting point.

Example: Bob leaves Chicago at 60 mph east, Susan leaves Pittsburgh (400 miles east of Chicago) at 40 mph west. Where do they meet?

Solution Outline:

• Let Bob's position: $x_B = 60t$

• Susan's position: $x_S = 400 - 40t$

• Set $x_B = x_S$ and solve for $t$ to find the meeting time and position.

Non-Uniform Motion and Acceleration

When an object's velocity changes, it experiences acceleration. The position-versus-time graph becomes curved, and the velocity-versus-time graph is no longer a horizontal line.

• Acceleration (a): The rate of change of velocity with respect to time.

• For constant acceleration, the position graph is a parabola, and the velocity graph is a straight line with nonzero slope.

Calculus in Kinematics: Derivatives and Integrals

Calculus provides tools to relate position, velocity, and acceleration:

• Velocity is the derivative of position: $v(t) = \frac{d x}{d t}$

• Acceleration is the derivative of velocity: $a(t) = \frac{d v}{d t}$

• Position is the integral of velocity: $x(t) = \int v(t) \, dt$

Graphical Interpretation: Slope and Area

The slope of the position-time graph gives velocity, while the area under the velocity-time graph gives displacement.

• Example: For $s = 2t^2$, the velocity is $v = 4t$.

• The area under the velocity-time graph between $t_1$ and $t_2$ equals the change in position.

Practice: Matching Graphs and Calculus Applications

Understanding how to match position and velocity graphs is essential for interpreting motion. Practice problems often ask which velocity graph corresponds to a given position graph, or vice versa.

Summary Table: Kinematic Quantities and Their Relationships

Quantity	Symbol	Definition	Graphical Meaning
Position	x	Location along a line	y-axis on position-time graph
Velocity	v	Rate of change of position	Slope of position-time graph
Acceleration	a	Rate of change of velocity	Slope of velocity-time graph
Displacement	\( \Delta x \)	Change in position	Area under velocity-time graph

Key Formulas for One-Dimensional Kinematics

• Uniform motion: $x = x_0 + v t$

• Constant acceleration: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity with constant acceleration: $v = v_0 + a t$

Additional info: These notes cover the essential calculus-based relationships between position, velocity, and acceleration in one-dimensional motion, including graphical interpretations and problem-solving strategies. Practice matching graphs and using calculus to move between position, velocity, and acceleration for a complete understanding of kinematics in one dimension.