Motion in One Dimension (Kinematics) – Study Notes

Motion in One Dimension

Introduction to Motion in Physics

Motion in physics is described using two main branches: kinematics and dynamics. Kinematics focuses on describing motion without considering its causes, while dynamics examines the forces and masses that produce motion.

• Kinematics: Describes motion in terms of displacement, velocity, and acceleration, without reference to forces.

• Dynamics: Explains motion by considering its causes, such as force, mass, and acceleration.

Assumptions in One-Dimensional Motion

• Motion occurs along a straight line (horizontal, vertical, or slanted).

• The moving object is treated as a point-like object (ignoring its size and shape).

• Only kinematics is considered; causes of motion (forces) are not discussed.

Position and Displacement

Definitions and Properties

Position refers to the location of an object along a coordinate axis. Displacement is the change in position and is a vector quantity, meaning it has both magnitude and direction.

• Displacement: $\Delta x = x_2 - x_1$

• Displacement can be positive or negative, depending on direction.

• The magnitude of displacement is always positive.

Example: If $x_1 = -5.0\,\text{m}$ and $x_2 = 2.0\,\text{m}$, then $\Delta x = 2.0 - (-5.0) = 7.0\,\text{m}$.

Graphs: Position as a Function of Time

Position-time graphs visually represent how an object's position changes over time. The slope of the graph at any point gives the velocity at that instant.

• Horizontal sections indicate the object is stationary.

• Steeper slopes indicate higher velocities.

Velocity and Speed

Average Velocity

Average velocity is defined as the total displacement divided by the total time taken.

• 

• Units: $\left[ v_{\text{vel}} \right] = \frac{\text{m}}{\text{s}} = \text{ms}^{-1}$

• On a position-time graph, average velocity is the slope of the line connecting two points.

Average Speed

Average speed is the total distance traveled divided by the total time taken. Unlike velocity, speed is always positive and does not depend on direction.

• $s_{\text{ave}} = \frac{\text{total distance}}{\Delta t}$

• Average speed can differ from average velocity if the path is not straight.

Instantaneous Velocity and Speed

Instantaneous velocity is the rate of change of position at a specific instant. It is the slope of the position-time graph at a point.

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

• Instantaneous speed is the magnitude of instantaneous velocity.

Calculus in Kinematics

Basic Derivatives and Integrals

Calculus is used to relate position, velocity, and acceleration. The derivative of position with respect to time gives velocity, and the derivative of velocity gives acceleration.

Function $f(x)$	Derivative $f'(x)$
$e^x$	$e^x$
$\sin(x)$	$\cos(x)$
$\cos(x)$	$-\sin(x)$
$x^n$	$n x^{n-1}$
$\ln(x)$	$1/x$
... (see full table for more)	... (see full table for more)

Indefinite Integrals:

Integral	Result
$\int k dx$	$kx + C$
$\int x^n dx$	$\frac{x^{n+1}}{n+1} + C$
$\int \sin x dx$	$-\cos x + C$
$\int \cos x dx$	$\sin x + C$
... (see full table for more)	... (see full table for more)

Acceleration

Average and Instantaneous Acceleration

Acceleration measures the rate of change of velocity. It can be average (over a time interval) or instantaneous (at a specific moment).

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

• Units: $\left[ a_{\text{ave}} \right] = \frac{\text{m}}{\text{s}^2} = \text{ms}^{-2}$

• Instantaneous acceleration: $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$

• On a velocity-time graph, acceleration is the slope at a point.

• If velocity and acceleration have the same sign, speed increases; if opposite, speed decreases.

Constant Acceleration

Basic Equations of Kinematics

For motion with constant acceleration, several key equations describe the relationships between position, velocity, acceleration, and time.

• $v = v_0 + at$

• $\Delta x = v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2a \Delta x$

These equations are derived using calculus and algebra, assuming acceleration $a$ is constant.

Graphical Interpretation

• The slope of the position-time graph gives velocity.

• The slope of the velocity-time graph gives acceleration.

• Areas under the velocity-time graph represent displacement.

• Areas under the acceleration-time graph represent change in velocity.

Graphical Integration

• $v - v_0 = \Delta v = \int_{t_0}^{t_1} a \, dt$

• $x - x_0 = \Delta x = \int_{t_0}^{t_1} v \, dt$

Free Fall: A Special Case of Constant Acceleration

Free Fall Equations

When an object is in free fall, it experiences constant acceleration due to gravity ($g = 9.8\,\text{m/s}^2$). The following equations apply, using $y$ for vertical displacement:

• $a = -g$

• $v = v_0 - gt$

• $\Delta y = v_0 t - \frac{1}{2} g t^2$

• $v^2 = v_0^2 - 2g \Delta y$

Example: A ball is tossed vertically upward with $v_0 = 12.0\,\text{m/s}$. Calculate time to reach maximum height, maximum height, total time to return, and velocity at the starting point.

• Time to maximum height: $t_{\text{max}} = \frac{v_0}{g}$

• Maximum height: $y = \frac{v_0^2}{2g}$

• Total time: $t_{\text{total}} = 2 t_{\text{max}}$

• Velocity at return: $v = -v_0$

Optional: Calculus Derivation of Kinematic Equations

Using calculus, the kinematic equations for constant acceleration can be derived:

• Start with $a = \frac{dv}{dt}$, integrate to get $v = v_0 + at$

• Integrate $v = \frac{dx}{dt}$ to get $x = x_0 + v_0 t + \frac{1}{2} a t^2$

Thus, $x - x_0 = v_0 t + \frac{1}{2} a t^2$

Summary Table: Kinematic Equations for Constant Acceleration

Equation	Description
$v = v_0 + at$	Velocity as a function of time
$\Delta x = v_0 t + \frac{1}{2} a t^2$	Displacement as a function of time
$v^2 = v_0^2 + 2a \Delta x$	Velocity as a function of displacement

Practice Problems and Examples

• Calculate the velocity and position of a particle given $x(t) = 4 - 27t + t^3$.

• Analyze the motion of a car with changing acceleration.

• Determine the speed and position of a falling object after a given time.

Additional info: These notes include calculus tables for derivatives and integrals to support the mathematical treatment of kinematics. Graphical analysis is emphasized for interpreting motion, velocity, and acceleration.