1. Motion in One and Two Dimensions

1.1 Motion with Constant Acceleration

Understanding motion with constant acceleration is fundamental in classical mechanics. This section introduces the equations of motion, velocity, and position for objects moving with constant acceleration, and explores their graphical interpretations.

• Definition: Constant acceleration means the velocity of an object changes by the same amount in each equal time interval.

• Key Equations:

  • Velocity as a function of time:

  • Position as a function of time:

  • Velocity-squared equation (no time):

  • Average velocity:

• Graphical Interpretation: The area under a velocity-time graph gives displacement; the slope of a position-time graph gives velocity.

• Example: A car decelerating from 110 km/h to 50 km/h over 5 seconds. Calculating acceleration and stopping distance using the above equations.

1.1.4 Freely Falling Objects

Objects in free fall experience constant acceleration due to gravity, denoted as (approximately near Earth's surface).

• Key Points:

  • Acceleration is downward and constant (ignoring air resistance).

  • Equations of motion apply with (if upward is positive).

• Example: A brick dropped from a building: is used to find the time to hit the ground and final velocity.

1.2 Motion with Non-Constant Acceleration

When acceleration varies with time, the standard equations do not apply. Instead, calculus is used to relate position, velocity, and acceleration.

• Key Equations:

  • Velocity from acceleration:

  • Position from velocity:

• Example: For , integrate to find and .

1.3 Motion in More Than One Dimension

Describing motion in two or three dimensions requires vectors for position, velocity, and acceleration.

• Position Vector:

• Displacement:

• Velocity:

• Acceleration:

• Magnitude of velocity (speed):

• Example: A squirrel's position as a function of time is given; find velocity and displacement at a specific time.

1.4 Projectile Motion

Projectile motion is a special case of two-dimensional motion where the only acceleration is due to gravity.

• Horizontal motion: (constant velocity)

• Vertical motion:

• Trajectory equation:

• Key Features:

  • Path is a parabola.

  • Horizontal and vertical motions are independent.

• Example: A diver jumps off a cliff; calculate time to hit the water and horizontal distance traveled.

1.5 Uniform Circular Motion

Uniform circular motion describes an object moving in a circle at constant speed. The direction of velocity changes, so there is always acceleration (centripetal acceleration) directed toward the center of the circle.

• Centripetal acceleration:

• Period of revolution:

• Alternate form:

• Example: Calculate the centripetal acceleration of a point on Earth's equator due to Earth's rotation.

1.6 Relative Velocity

Relative velocity describes how the velocity of an object appears different to observers in different frames of reference.

• One Dimension:

• Two or Three Dimensions:

• Galilean Transformation: Used for non-relativistic speeds to relate velocities in different frames.

• Example: A person walking inside a moving train; calculate their velocity relative to the ground and to the train.

Summary Table: Equations of Motion for Constant Acceleration

Additional info:

• This summary covers the first major topics in a college-level Physics 1 course, including kinematics in one and two dimensions, projectile and circular motion, and relative velocity. The notes are based on the structure and content of the provided lecture notes, with expanded academic context and examples for clarity.