Kinematics in One Dimension: Position, Velocity, and Acceleration

Chapter 2 – Kinematics in 1 Dimension

Introduction and Assumptions

Kinematics in one dimension studies the motion of objects along a straight line, focusing on position, velocity, and acceleration. Two simplifying assumptions are made:

• Particle Model: The moving object is treated as a point-like particle, ignoring rotation.

• Straight-Line Motion: The motion considered is strictly along a straight line (horizontal or vertical).

Physical Quantities in Kinematics

Physical quantities are classified as vectors or scalars:

• Vectors: Displacement, velocity, acceleration (direction indicated by sign in 1D).

• Scalars: Distance travelled, speed.

Position and Displacement

Position and displacement are fundamental concepts in describing motion:

• Position: Measured by x (horizontal) or y (vertical) relative to an origin.

• Displacement: The change in position, given by $\Delta x = x_2 - x_1$.

• Displacement is a vector quantity and may differ from the distance travelled.

• Motion is relative—depends on the chosen reference frame.

Average Velocity and Speed

Velocity and speed quantify how fast and in what direction an object moves:

• Average Velocity (vector): $v_{ave} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

• Average Speed (scalar): $v_{ave} = \frac{\text{distance travelled}}{\text{elapsed time}}$

• Average speed is not the magnitude of average velocity unless the path is straight with no reversals.

Example: From Pillar to Post

Suppose you run 200 m east at 5.0 m/s, then 280 m west at 4.0 m/s. Calculate:

• Distance travelled: $200\,\text{m} + 280\,\text{m} = 480\,\text{m}$

• Displacement: $x_f - x_i = -80\,\text{m}$

• Average speed: $v_{ave} = \frac{480\,\text{m}}{110\,\text{s}} = 4.4\,\text{m/s}$

• Average velocity: $v_{ave} = \frac{-80\,\text{m}}{110\,\text{s}} = -0.73\,\text{m/s}$

Graphical Representation of Motion

Graphs are essential for visualizing motion:

• x-t Graph: Position vs. time; slope gives velocity.

• Slope of the line: $v_{ave} = \frac{\Delta x}{\Delta t}$

Instantaneous Velocity

Instantaneous velocity describes the rate of change of position at a specific instant:

• $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• It is the slope of the tangent to the x-t curve at a given time.

• The sign of v indicates direction.

Velocity and Slope of the Tangent

• Positive slope: Positive velocity.

• Negative slope: Negative velocity.

Differentiation in Kinematics

Differentiation is used to find velocity and acceleration from position functions:

• If $x = bt^m$, then $v = \frac{dx}{dt} = mbt^{m-1}$

• $a = \frac{dv}{dt} = (m-1)mbt^{m-2}$

• Example: $x(t) = 5 + 4t + 3t^2 \rightarrow v(t) = 4 + 6t$

Acceleration: Average and Instantaneous

Acceleration measures the rate of change of velocity:

• Average acceleration: $a_{ave} = \frac{\Delta v}{\Delta t} = \frac{v_2 - v_1}{t_2 - t_1}$

• Instantaneous acceleration: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Second Derivative Interpretation

• $a = \frac{dv}{dt} = \frac{d}{dt} \left( \frac{dx}{dt} \right ) = \frac{d^2x}{dt^2}$

• Example: $x(t) = 5 + 4t + 3t^2 \rightarrow v(t) = 4 + 6t \rightarrow a(t) = 6$

Equations of Motion

Equations of motion relate position, velocity, acceleration, and time:

• General equation: $x_2 = x_1 + v_{ave} t$

• For constant velocity: $x_2 = x_1 + vt$

Motion with Constant Acceleration

When acceleration is constant, the following equations apply:

• Velocity: $v(t) = v_1 + at$

• Position: $x_2 = x_1 + v_1 t + \frac{1}{2} a t^2$

• Galilei’s formula: $v^2 = v_0^2 + 2a(x - x_0)$

• Alternate forms are used depending on which variable is missing.

The Five Kinematic Equations

Equation	Missing Variable
$v = v_0 + at$	x
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	v
$v^2 = v_0^2 + 2a(x - x_0)$	t
$x = x_0 + \frac{1}{2}(v_0 + v)t$	a
$x = x_0 + vt - \frac{1}{2} a t^2$	$v_0$

Worked Examples

Example 1: Antelope Motion

An antelope moving with constant acceleration covers 70.0 m in 7.00 s. Its speed at the second point is 15.0 m/s. Find its speed at the first point and its acceleration.

Example 2: Solar Car Acceleration

A solar-powered car accelerates from rest, cruises, and then brakes to a stop. Compute average and instantaneous acceleration over specified intervals using the velocity-time graph.

Summary Table: Vectors vs. Scalars in 1D Kinematics

Quantity	Type	Definition
Displacement	Vector	$\Delta x = x_2 - x_1$
Distance travelled	Scalar	Total path length
Velocity	Vector	$v = \frac{dx}{dt}$
Speed	Scalar	$v_{ave} = \frac{\text{distance}}{\text{time}}$
Acceleration	Vector	$a = \frac{dv}{dt}$

Additional info: These notes cover the foundational concepts of kinematics in one dimension, including graphical analysis, differentiation, and the use of kinematic equations for problem solving. The examples provided illustrate practical applications of these principles in real-world scenarios.