(CH3) Kinematics in Two Dimensions: Position, Velocity, and Acceleration

Kinematics in Two Dimensions

Introduction

Kinematics in two dimensions extends the study of motion to include vector quantities such as position, velocity, and acceleration. This topic is fundamental in understanding projectile motion, uniform circular motion, and relative motion.

• Position, velocity, and acceleration are treated as vectors.

• Special cases include projectile motion and uniform circular motion.

• Relative motion is also considered.

Position and Displacement

Definitions and Vector Representation

The position of an object is described by its position vector \( \vec{r} \). The displacement is the change in position:

• Displacement: The vector difference between the final and initial position vectors.

$\Delta \vec{r} = \vec{r}_f - \vec{r}_i$

• Displacement is a vector quantity, having both magnitude and direction.

• It is independent of the path taken between two points.

General Motion Ideas

Vector Notation in Multi-Dimensional Motion

In two- or three-dimensional kinematics, the principles are similar to one-dimensional motion, but full vector notation is required.

• Positive and negative signs alone are insufficient; direction must be specified using vectors.

Average Velocity

Definition and Properties

Average velocity is defined as the displacement divided by the time interval over which the displacement occurs:

$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$

• The direction of average velocity is the same as the direction of displacement.

• Average velocity is independent of the path taken between two points.

Instantaneous Velocity

Definition and Graphical Interpretation

Instantaneous velocity is the limit of the average velocity as the time interval approaches zero:

$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

• The direction of instantaneous velocity is tangent to the path at a given point and in the direction of motion.

• The magnitude of the instantaneous velocity is called speed, which is a scalar quantity.

Average Acceleration

Definition and Vector Nature

Average acceleration is the change in the instantaneous velocity vector divided by the time interval during which the change occurs:

$\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i}$

• The direction of average acceleration is along the change in velocity vector, \( \Delta \vec{v} \).

Instantaneous Acceleration

Definition and Calculation

Instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero:

$\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt}$

• Instantaneous acceleration is the derivative of the velocity vector with respect to time.

Producing an Acceleration

Causes of Acceleration

Acceleration can result from changes in either the magnitude or direction of the velocity vector, or both.

• The magnitude of velocity may change (speeding up or slowing down).

• The direction of velocity may change (turning or curving motion).

• Both magnitude and direction may change simultaneously.

Kinematic Equations for Two-Dimensional Motion

Equations and Independence of Motion

When acceleration is constant, the motion in two dimensions can be described by equations similar to those in one dimension, applied independently to each axis.

• Two-dimensional motion can be modeled as two independent motions along perpendicular axes (x and y).

• Acceleration in one direction does not affect motion in the perpendicular direction.

Key Equations:

• Position as a function of time: $\vec{r}(t) = \vec{r}_i + \vec{v}_i t + \frac{1}{2} \vec{a} t^2$

• Velocity as a function of time: $\vec{v}(t) = \vec{v}_i + \vec{a} t$

Example: If a projectile is launched with an initial velocity \( \vec{v}_i \) at an angle, its motion can be analyzed by resolving the velocity and acceleration into x and y components and applying the above equations to each direction independently.