Study Notes on One Dimensional Kinematics

Chapter 4: One Dimensional Kinematics

4.1 Introduction

• Kinematics is the study of motion, which comes from the Greek word "kinema" meaning movement.

• Analyzing motion in physics can be complex, especially when considering time and space, as shown in relativity.

• A coordinate system (reference frame) is needed to describe motion accurately.

• Important concepts like position, velocity, and acceleration come into play once we have a reference frame.

• A one-dimensional Cartesian coordinate system uses a unit vector $\boldsymbol{i}$ to represent the direction of increasing x-coordinate.

4.2 Position, Time Interval, Displacement

4.2.1 Position

• Position is the location of an object's center of mass relative to a fixed point (the origin) and is denoted as $x(t)$.

• It can be positive (to the right of the origin), zero (at the origin), or negative (to the left of the origin).

• Position is a vector, meaning it has both a direction and a magnitude:
    $\boldsymbol{x}(t) = x(t) \boldsymbol{i}.$ The SI unit is meter (m). The initial position at time $t = 0$ is denoted as $x_0 ext{ if } x(t=0).$

4.2.2 Time Interval

• A time interval, labeled $[t_1, t_2]$, is the difference between two times:
    $riangle t = t_2 - t_1.$ The SI unit is seconds (s).

4.2.3 Displacement

• Displacement refers to the change in position of an object from time $t_1$ to $t_2$:
    $riangle \boldsymbol{x} = x(t_2) - x(t_1) \boldsymbol{i}.$

• Displacement is also a vector quantity.

4.3 Velocity

• The terms "speed" and "velocity" have specific definitions in mathematics despite their general usage.

• We define average quantities over a time interval, and then we look at instantaneous values.

• Instantaneous velocity is derived from the position function as the rate of change of position with respect to time.

4.3.1 Average Velocity

• The average velocity during a time interval $riangle t$ is:
    $v_x = rac{ riangle x}{ riangle t}.$ The SI unit is meters per second (m/s).

4.3.2 Instantaneous Velocity

• For a constantly moving object, the average velocity over an interval resembles the slope between two points on a curve:
    $ext{Slope} = rac{ ext{Change in position}}{ ext{Change in time}}.$ As $riangle t$ gets smaller, the average velocity gets closer to the slope of the tangent line to the curve at time $t$.

• Instantaneous velocity is therefore defined as:
    $\boldsymbol{v}(t) = \boldsymbol{v}_x(t) \boldsymbol{i}.$

4.4 Acceleration

• Acceleration is defined as the change in velocity over a time interval.

4.4.1 Average Acceleration

• Average acceleration can be calculated as:
    $riangle v = v(t + riangle t) - v(t).$ The average acceleration vector is:
    $\boldsymbol{a}(t) = rac{ riangle \boldsymbol{v}}{ riangle t} \boldsymbol{i}.$ The SI unit is meters per second squared (m/s²).

4.4.2 Instantaneous Acceleration

• Instantaneous acceleration is defined similarly to instantaneous velocity:
    $a_x(t) = rac{dv_x(t)}{dt}.$

Example

• Detailed examples will help apply these concepts, showing calculations and interpretations of motion in various scenarios.

4.5 Constant Acceleration

• Constant acceleration can be visualized as areas under graphs representing velocity and time.

4.6 Non-Constant Acceleration

• When acceleration varies, we use integrals and areas to describe how velocity changes over time.

• Key equations relate position, velocity, and acceleration, forming the foundation for understanding motion under different acceleration conditions.