Kinematics in One Dimension

Displacement, Velocity, and Acceleration

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In one-dimensional kinematics, we focus on motion along a straight line, using key quantities such as displacement, velocity, and acceleration.

• Displacement (\( \Delta x \)): The change in position of an object. It is a vector quantity, meaning it has both magnitude and direction. where \( x_f \) is the final position and \( x_i \) is the initial position.

• Average Velocity (\( v_{avg} \)): The total displacement divided by the total time taken.

• Instantaneous Velocity (\( v \)): The velocity of an object at a specific instant. It is the derivative of position with respect to time.

• Average Acceleration (\( a_{avg} \)): The change in velocity divided by the time interval.

• Instantaneous Acceleration (\( a \)): The acceleration at a specific instant. It is the derivative of velocity with respect to time.

Equations of Motion for Constant Acceleration

When acceleration is constant, the following kinematic equations are used to solve problems involving displacement, velocity, and time:

Where:

• \( x_0 \): Initial position

• \( v_0 \): Initial velocity

• \( a \): Constant acceleration

• \( t \): Time elapsed

Example: Free Fall

When an object is dropped from rest, it accelerates downward due to gravity (\( g \approx 9.8\, \text{m/s}^2 \)). The equations above can be applied with \( a = -g \) (if upward is positive).

• Example Problem: An object is dropped from a height of 20 m. How long does it take to reach the ground?

• Solution: Use with m, , , m/s.

• Solving for gives s.

Summary Table: Kinematic Quantities

Additional info: The notes also mention the importance of sign conventions (positive and negative directions) and the use of derivatives for instantaneous quantities, which are foundational for calculus-based physics.