Quantities of 2D Motion and Projectile Motion

Review of 1D Motion

Before exploring two-dimensional motion, it is important to recall the equations of motion for one-dimensional (1D) systems, both with constant velocity and constant acceleration.

• Constant Velocity (1D): The position as a function of time is given by:

• Constant Acceleration (1D): The position and velocity as functions of time are:

Free Fall in 1D

Free fall describes motion under the influence of gravity alone. For objects near Earth's surface, the acceleration due to gravity is approximately downward, denoted as .

• Gravitational Acceleration: (downward)

• Equations of Free Fall (vertical motion):

• Example: A ball is thrown upward from a building. Use the above equations to find maximum height and time to hit the ground.

2D Motion: Vectors and Kinematics

Two-dimensional motion requires the use of vectors to describe position, velocity, and acceleration. The same kinematic principles apply as in 1D, but each vector quantity has components in both the x and y directions.

Position Vector,

• Definition: The position vector locates an object in space relative to the origin.

• Displacement:

Velocity

• Average Velocity:

• Instantaneous Velocity:

Acceleration

• Average Acceleration:

• Instantaneous Acceleration:

Constant Acceleration in 2D and 3D

• Each component follows the 1D kinematic equations:

• Vector Notation:

Worked Example

A girl runs in a park with position , , . Find her velocity and acceleration at .

• Velocity:

• Acceleration:

Projectile Motion

Projectile motion describes the curved path of an object launched near Earth's surface, moving under the influence of gravity alone (air resistance is neglected).

• Key Features:

  • Constant downward acceleration ()

  • Parabolic trajectory

  • Horizontal and vertical motions are independent

Horizontal and Vertical Motion

• Horizontal motion: constant velocity ()

• Vertical motion: constant acceleration ()

• Equations:

• Example: An object dropped from an airplane moves horizontally with the plane while falling vertically under gravity.

Projectile Motion Diagram and Equations

Symmetric Projectile Motion

• Range (): The horizontal distance traveled by a projectile launched and landing at the same height.

• Time to reach maximum height equals time to descend from maximum height.

Sample Problems and Applications

• Finding maximum height, range, and time of flight for projectiles.

• Solving for unknowns using quadratic equations when necessary.

• Comparing speeds and accelerations at different points in the trajectory.

Conceptual Questions (iClicker Examples)

• At what point is the speed of a projectile minimum? At the highest point of the trajectory.

• At what point is the acceleration of a projectile greatest? Acceleration is constant and equal to at all points (ignoring air resistance).

• Comparing time of flight for projectiles launched at different angles but with the same speed.

Summary Table: Key Equations for Projectile Motion

Next Topics

• Further exploration of projectile motion

• Introduction to circular motion

Additional info: These notes are based on slides for a university-level introductory physics course (mechanics), focusing on kinematics in one and two dimensions, with emphasis on projectile motion and vector analysis.