Kinematics: Motion in One Dimension (Physics 1, Chapter 2)

Kinematics: Motion in One Dimension

Instantaneous Velocity

Instantaneous velocity describes how fast and in what direction an object is moving at a specific instant in time. It is a fundamental concept in kinematics, especially when analyzing non-uniform motion.

• Definition: The instantaneous velocity at time t is the rate of change of position with respect to time at that instant.

• Graphical Interpretation: On a position-versus-time graph, the instantaneous velocity at time t is the slope of the tangent to the curve at that point.

• Mathematical Expression: $v_s = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} = \frac{ds}{dt}$ where $s$ is position and $t$ is time.

• Comparison to Average Velocity: The average velocity over a time interval $\Delta t$ is $\frac{\Delta s}{\Delta t}$, but as $\Delta t$ becomes very small, this approaches the instantaneous velocity.

• Example: If a car's position changes non-linearly over time, the instantaneous velocity at any moment is found by taking the derivative of its position function with respect to time.

Average Acceleration

Average acceleration quantifies how quickly an object's velocity changes over a time interval. It is a vector quantity, meaning it has both magnitude and direction.

• Definition: The average acceleration during a time interval $\Delta t$ is the change in velocity divided by the time interval.

• SI Units: The units are meters per second squared (m/s2).

• Mathematical Expression: $a_{avg} = \frac{\Delta v_s}{\Delta t}$ where $\Delta v_s$ is the change in velocity and $\Delta t$ is the time interval.

• Graphical Interpretation: On a velocity-versus-time graph, the average acceleration is the slope of the straight line connecting two points.

• Constant Acceleration: If acceleration is constant, instantaneous acceleration $a_s$ equals average acceleration $a_{avg}$.

• Vector Nature: Acceleration, like velocity, is a vector and can be positive (speeding up in the positive direction) or negative (slowing down or speeding up in the negative direction).

• Example: If a car's velocity increases from 0 m/s to 20 m/s in 4 seconds, its average acceleration is $\frac{20\,\text{m/s} - 0\,\text{m/s}}{4\,\text{s}} = 5\,\text{m/s}^2$.

Graphical Representation of Motion

Understanding motion often involves interpreting graphs of position, velocity, and acceleration versus time.

• Position-Time Graph: The slope at any point gives the instantaneous velocity.

• Velocity-Time Graph: The slope at any point gives the instantaneous acceleration.

• Area Under Curve: The area under a velocity-time graph between two times gives the displacement.

• Example: For a car accelerating uniformly, the position-time graph is a curve, the velocity-time graph is a straight line, and the acceleration-time graph is a horizontal line.

Summary Table: Key Kinematic Quantities

Quantity	Definition	Graphical Meaning	Formula
Instantaneous Velocity	Rate of change of position at a specific time	Slope of position-time graph	$v_s = \frac{ds}{dt}$
Average Velocity	Change in position over time interval	Slope of chord on position-time graph	$v_{avg} = \frac{\Delta s}{\Delta t}$
Instantaneous Acceleration	Rate of change of velocity at a specific time	Slope of velocity-time graph	$a_s = \frac{dv}{dt}$
Average Acceleration	Change in velocity over time interval	Slope of chord on velocity-time graph	$a_{avg} = \frac{\Delta v}{\Delta t}$

Additional info: These notes expand on the provided slides by including definitions, formulas, and graphical interpretations for each concept, ensuring a self-contained study guide for kinematics in one dimension.