Kinematics in Multiple Dimensions and Relative Velocity: Study Notes

Kinematics in 1-2 Dimensions & Relative Velocity

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. In college-level physics, kinematics is often studied in one and two dimensions, which allows for the analysis of more complex motion such as projectile motion and relative velocity.

• Displacement: The change in position of an object.

• Velocity: The rate of change of displacement with respect to time.

• Acceleration: The rate of change of velocity with respect to time.

Example: A car moving along a straight road at a constant speed is an example of one-dimensional kinematics.

Kinematics in Two Dimensions

When motion occurs in a plane, both the x (horizontal) and y (vertical) components must be considered. This is essential for analyzing projectile motion and objects moving in more than one direction.

• Projectile Motion: The motion of an object thrown or projected into the air, subject only to gravity.

• Trajectory: The path followed by a projectile, typically a parabola under uniform gravity.

Equations for Projectile Motion:

• Horizontal motion (no acceleration): $x = x_0 + v_{x,0} t$

• Vertical motion (constant acceleration due to gravity): $y = y_0 + v_{y,0} t - \frac{1}{2} g t^2$ $v_{y} = v_{y,0} - g t$

Example: A ball launched upward from a moving cart will follow a parabolic trajectory, and its motion must be analyzed in both x and y directions.

Frames of Reference and Relative Velocity

Frames of Reference

A frame of reference is a coordinate system used to measure the position, velocity, and acceleration of objects. The choice of frame affects how motion is described.

• Stationary Frame: Often called the lab frame, where the observer is at rest.

• Moving Frame: A frame that moves at a constant velocity relative to the stationary frame.

Example: Observing a person walking inside a moving train; the train is the moving frame, and the ground is the stationary frame.

Galilean Transformation of Velocity

When analyzing motion from different frames of reference, the Galilean transformation is used to relate velocities measured in each frame. This transformation is valid only when the frames move at constant velocity relative to each other (i.e., non-accelerating frames).

• Galilean Transformation Equation: $\vec{v} = \vec{w} + \vec{v}_F$ Where: $\vec{v}$ = velocity in the stationary (lab) frame $\vec{w}$ = velocity in the moving frame $\vec{v}_F$ = velocity of the moving frame relative to the stationary frame

• Relative Velocity: $\vec{w} = \vec{v} - \vec{v}_F$ This gives the velocity as measured in the moving frame.

Example: If a train moves at 30 m/s and a person walks forward inside the train at 2 m/s, the person's velocity relative to the ground is $32$ m/s.

Applicability of Galilean Transformations

Galilean transformations are only valid for frames moving at constant velocity relative to each other. If the frames are accelerating, more complex transformations are required, and fictitious forces may need to be considered.

• Constant Velocity: Use Galilean transformation.

• Accelerating Frames: Galilean transformation does not apply; fictitious forces (e.g., centrifugal force) may appear.

Example: In a turning car (accelerating frame), you feel pushed outward due to a fictitious force.

Projectile Motion in Moving Frames

Analyzing Projectile Motion from a Moving Cart

When a projectile is launched from a moving cart, its initial velocity must be considered in both the x and y directions, and relative to both the cart and the ground.

• Initial Conditions: The initial velocity of the projectile is the vector sum of the cart's velocity and the velocity imparted by the launch.

• Trajectory: The projectile will land ahead, behind, or in the cart depending on the initial velocity components.

Example: If a ball is launched upward from a cart moving at velocity $v_{cart}$, the ball's initial velocity in the x-direction is $v_{cart}$, and in the y-direction is the launch velocity $v_{y,0}$.

Practice Problems and Applications

Relative Velocity Practice

Consider a train moving at 30 m/s and a person walking toward the front at 2 m/s. The velocity of the person relative to the ground is:

• $v_{person,ground} = v_{train} + v_{person,train} = 30 + 2 = 32$ m/s

If two balls are thrown in opposite directions from a moving cart, their velocities relative to the ground and the cart must be calculated using the Galilean transformation.

Help Lab Schedule

Physics Help Lab Hours

Physics help labs are available throughout the week for students to get assistance with homework, quizzes, and exam preparation. The schedule is organized by day, time, and location.

Day	Time	Location
Monday	Various times	CSC 205, SFEBB 130
Tuesday	Various times	CSC 205, SFEBB 130
Wednesday	4:30pm-6:30pm	CSC 205 (2nd floor Crocker)
Thursday	3:30pm-7pm	SFEBB 130 (basement Eccles Business)

Additional info: Students are encouraged to attend help labs and discussion sections for collaborative learning and support.