Kinematics: Motion Along a Straight Line

Kinematics: Motion Along a Straight Line

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In this section, we focus on motion along a single axis (the s-axis), analyzing position, velocity, and acceleration as functions of time.

• Position (s): The location of an object along a chosen axis, typically measured in meters (m).

• Velocity (v): The rate of change of position with respect to time. It can be positive or negative, indicating direction.

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

Key Definitions and Concepts

• Displacement (Δs): The change in position of an object: $\Delta s = s_f - s_i$

• Average Velocity (\bar{v}): The total displacement divided by the total time interval: $\bar{v} = \frac{\Delta s}{\Delta t}$

• Instantaneous Velocity (v): The velocity at a specific instant, defined as the derivative of position with respect to time: $v = \frac{ds}{dt}$

• Average Acceleration (\bar{a}): The change in velocity divided by the time interval: $\bar{a} = \frac{\Delta v}{\Delta t}$

• Instantaneous Acceleration (a): The acceleration at a specific instant, defined as the derivative of velocity with respect to time: $a = \frac{dv}{dt}$

Graphical Interpretation of Motion

Graphs are essential tools for visualizing motion. The most common graphs in kinematics are:

• Position vs. Time (s-t) Graph: Shows how position changes over time. The slope at any point gives the instantaneous velocity.

• Velocity vs. Time (v-t) Graph: Shows how velocity changes over time. The slope at any point gives the instantaneous acceleration. The area under the curve gives the displacement.

• Acceleration vs. Time (a-t) Graph: Shows how acceleration changes over time. The area under the curve gives the change in velocity.

Equations of Motion for Constant Acceleration

When acceleration is constant, the following kinematic equations apply:

• $v = v_0 + a t$

• $s = s_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2a(s - s_0)$

• $\bar{v} = \frac{v_0 + v}{2}$ (for constant acceleration)

Where:

• $s_0$ = initial position

• $v_0$ = initial velocity

• $a$ = constant acceleration

• $t$ = time elapsed

Types of Motion

• Uniform Motion: Motion with constant velocity (zero acceleration). The position-time graph is a straight line.

• Uniformly Accelerated Motion: Motion with constant acceleration. The position-time graph is a parabola, and the velocity-time graph is a straight line.

Example: Calculating Position from Velocity

If an object starts from rest ($v_0 = 0$) and accelerates at $a = 2\,\mathrm{m/s^2}$ for $t = 5\,\mathrm{s}$, its position is:

• $s = s_0 + v_0 t + \frac{1}{2} a t^2$

• $s = 0 + 0 + \frac{1}{2} \times 2 \times (5)^2 = 25\,\mathrm{m}$

Table: Summary of Kinematic Quantities

Quantity	Symbol	Definition	SI Unit
Position	s	Location along axis	meter (m)
Displacement	$\Delta s$	Change in position	meter (m)
Velocity	v	Rate of change of position	meter/second (m/s)
Acceleration	a	Rate of change of velocity	meter/second2 (m/s2)

Additional info:

• Some content was inferred and clarified based on standard introductory physics curriculum, as the original notes were fragmented and partially illegible.

• References to 'Power law function' and 'Whiteboard problem' suggest practice problems and more advanced function types, but details were not provided.