1D Kinematics: Study Notes for College Physics

1D Kinematics

Introduction to Mechanics

Mechanics is the branch of physics concerned with the study of motion and the forces that cause it. Mechanics is divided into two main areas: Kinematics and Dynamics.

• Kinematics: Describes how objects move (position, velocity, acceleration).

• Dynamics: Explains why objects move (forces and interactions).

Speed

Speed is a scalar quantity that measures how fast an object is moving, regardless of direction.

• Definition: Speed is the distance traveled divided by the time taken.

• Formula: $s = \frac{d}{t}$ where $s$ is speed, $d$ is distance, and $t$ is time.

• Application: You can plot distance against time in a scatterplot to visualize speed.

Displacement

Displacement is a vector quantity representing the change in position of an object.

• Definition: Displacement is the straight-line distance from the initial to the final position, including direction.

• Example: If you walk 3 meters east and then 4 meters west, your displacement is 1 meter west.

Average Speed and Average Velocity

When an object's speed changes, we calculate its average speed and average velocity.

• Average Speed: $\bar{s} = \frac{d}{t}$ (scalar)

• Average Velocity: $\bar{v} = \frac{\Delta x}{\Delta t}$ where $\Delta x$ is displacement and $\Delta t$ is time interval.

Velocity

Velocity is a vector quantity describing the rate of change of position with respect to time, including direction.

• Formula: $v = \frac{\Delta x}{\Delta t}$

• Instantaneous Velocity: The velocity of an object at a specific moment in time. $v_{inst} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

• Example: If a car moves 10 meters east in 2 seconds, its average velocity is $5\ \text{m/s}$ east.

Acceleration

Acceleration is the rate of change of velocity with respect to time.

• Formula: $a = \frac{\Delta v}{\Delta t}$

• Instantaneous Acceleration: $a_{inst} = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}$

• Average Acceleration: $a_{avg} = \frac{v_f - v_0}{t}$ where $v_f$ is final velocity, $v_0$ is initial velocity, and $t$ is time interval.

• Velocity and Acceleration Direction:

  • If velocity and acceleration are in the same direction, the object speeds up.

  • If they are in opposite directions, the object slows down.

Kinematic Equations for Constant Acceleration

For motion with constant acceleration, the following kinematic equations are used:

• Velocity as a function of time: $v = v_0 + at$

• Displacement as a function of time: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Velocity as a function of displacement: $v^2 = v_0^2 + 2a(x - x_0)$

• Average velocity: $\bar{v} = \frac{v_0 + v_f}{2}$

Graphical Analysis

Graphs are useful for visualizing motion:

• Position vs. Time: Slope gives velocity.

• Velocity vs. Time: Slope gives acceleration; area under the curve gives displacement.

• Example: For a velocity-time graph with a straight line, the acceleration is constant and can be calculated from the slope.

Worked Example: Velocity-Time Graph

Given a velocity-time graph:

• Calculate acceleration: $a = \frac{v_2 - v_1}{t_2 - t_1}$

• Write the equation for velocity: $v = 3t + 3$

Area Under Velocity-Time Graph

The area under a velocity-time graph represents the displacement of the object.

• Example: If the area under the graph from $t=0$ to $t=T$ is $A$, the velocity at time $T$ can be found using: $v = \frac{2A}{T} - v_0$

Summary Table: Kinematic Quantities

Quantity	Definition	Formula	Type
Speed	Distance per unit time	$s = \frac{d}{t}$	Scalar
Velocity	Displacement per unit time	$v = \frac{\Delta x}{\Delta t}$	Vector
Acceleration	Change in velocity per unit time	$a = \frac{\Delta v}{\Delta t}$	Vector

Additional info:

• For free-fall motion, acceleration due to gravity is $a = -g = -9.81\ \text{m/s}^2$.

• In 1D kinematics, direction is indicated by sign (positive or negative).

• Instantaneous quantities require calculus for precise calculation.