Kinematics in Two Dimensions: Lecture 7 Notes

Date/Time: Wednesday, October 1, 2025, $2:15-3:10$ pm (class period).
Location: PHY-001 (the normal classroom, Physics Lecture Hall on Busch Campus). All students will take the exam at this location.
Arrival Time: Students should plan to arrive no later than $2:00$ pm. No extra time will be given for late arrivals.
Materials to Bring: Pencil/pen, a calculator, and Rutgers Student ID.
Material Covered: Chapters $1-3$ and related lectures, recitations/workshops/quizzes, and homeworks.
Format: $10$ multiple-choice questions.
Duration: $55$ -minute class period.
Rules: Closed book/note. An equation sheet will be provided within the exam.
Related Resources: Exam information, Exam $1$ Announcement, Exam $1$ Important Information Page, Practice Exam $1$ , and Equation Sheet for Exam $1$ are available on Canvas.
Lecture 8: Will be an exam review, focusing on problems from the practice exam.

Read Chapters $1-3$ of the textbook and work on additional problems at the end of each chapter.
Review lecture slides provided in Canvas weekly.
Refer back to relevant homeworks, recitation/workshop materials, and quizzes to review work and address any past difficulties.
Work on the provided practice exam using only the provided equation sheet. The practice exam's cover page will mimic the actual exam.
Attend instructors' office hours for guidance.

This lecture covers solving problems related to motion in a plane.
Key topics include Projectile Motion and Relative Motion.
Relevance: Linear motion introduces basic concepts, but most real-world motion occurs in two or three dimensions (e.g., curved trajectories of balls, cars turning, planetary orbits, electron spirals in magnetic fields). Two-dimensional motion is ubiquitous.

Displacement: A vector, denoted as $\Delta \vec{r} = \vec{r}f - \vec{r}i$ .
Velocity and Acceleration: Also vectors, expressible in component form:
- Velocity: $\vec{v} = \vec{v}x + \vec{v}y + \vec{v}_z$
- Acceleration: $\vec{a} = \vec{a}x + \vec{a}y + \vec{a}_z$
Problem Solving Strategy: Because displacement, velocity, and acceleration are vectors, any $2$ (or $3$ ) dimensional problem can be solved by treating it as two (or three) $1$ -dimensional problems in perpendicular directions (e.g., x and y).

A particle moving along a curved path in the $xy$ -plane has its position located by a position vector $\vec{r}$ .
A graph of $y$ versus $x$ provides an actual picture of the trajectory, not an abstract representation of motion.

Definition: Two-dimensional free-fall motion under the exclusive influence of gravity.
Projectile: An object moving in two dimensions solely under Earth's gravity.
Trajectory: Follows a parabolic path, assuming air resistance is negligible.
Motion Characteristics:
- Horizontal Direction: Uniform motion ( $a_x = 0$ ).
- Vertical Direction: Constant acceleration, $\vec{a}_y = -g$ . Projectiles are in free-fall.

To demonstrate parabolic trajectory, $y$ is written as a function of $x$ . The resulting form is: $y = ax + bx^2$ which is the equation of a parabola.

Projectile motion can be understood by analyzing the horizontal and vertical motions independently.

Scenario: From the same height and at the same time, one ball is dropped, and another is fired horizontally.
Question: Which ball will hit the ground first?
Answer: They both hit at the same time.
Reasoning: The horizontal motion does not affect the vertical motion. Both balls experience the same constant downward acceleration due to gravity ( $g$ ) and start with zero initial vertical velocity (assuming the fired ball is launched purely horizontally). Therefore, their vertical motion, and thus the time to hit the ground, is identical.
- This concept is often demonstrated with a