Kinematics in 1D – PHYSICS 1051 Study Notes

Kinematics in 1D

Introduction to 1D Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In one-dimensional (1D) kinematics, we simplify the world by modeling all motion as occurring along a straight line, allowing us to focus on the essential features of motion.

• Point Particle Model: Objects are treated as if all their mass is concentrated at a single point.

• 1D Assumption: Motion is restricted to a straight line, neglecting other dimensions for simplicity.

• Example: A train moving along a straight track can be modeled as a point particle in 1D.

Key Concepts in Kinematics

Geometry of Motion

Kinematics involves the mathematical description of the geometry of motion. The main quantities of interest are:

• Position (x): The location of an object along a straight line, measured in meters (m).

• Displacement (Δx): The change in position, defined as final position minus initial position.

• Speed: The rate at which distance is covered, regardless of direction.

• Velocity (v): The rate of change of position, including direction.

• Acceleration (a): The rate at which velocity changes with time.

All these quantities are measured using combinations of space (meters) and time (seconds).

Distance vs. Displacement

It is important to distinguish between distance and displacement:

• Distance: The total length of the path traveled, regardless of direction (always positive).

• Displacement: The net change in position, which can be positive, negative, or zero.

• Example: An ant crawls from x = 40 cm to x = 10 cm, then back to x = 40 cm. Total distance = 60 cm, total displacement = 0 cm.

Average Speed vs. Average Velocity

Average speed and average velocity are related but distinct concepts:

• Average Speed: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

• Average Velocity: $\text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}}$

• Example: If the ant takes 10 s to go from x = 40 cm to x = 10 cm, and another 10 s to return, average speed = 3 cm/s, average velocity = 0 cm/s.

Graphical Analysis in 1D Kinematics

Position vs. Time Graphs

Graphs are a powerful tool for visualizing motion in 1D:

• Position-Time Graph: Plots position (y-axis) versus time (x-axis).

• Slope: The slope of the position-time graph at any point gives the instantaneous velocity.

• Straight Line: Indicates uniform (constant) velocity.

• Curved Line: Indicates changing velocity (acceleration).

Velocity vs. Time Graphs

• Velocity-Time Graph: Plots velocity (y-axis) versus time (x-axis).

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

• Area Under Curve: The area under the velocity-time graph gives the displacement.

Mathematical Tools: Derivatives and Integrals

Derivatives in Kinematics

Derivatives are used to find instantaneous rates of change:

• Instantaneous Velocity: $v = \frac{dx}{dt}$

• Instantaneous Acceleration: $a = \frac{dv}{dt}$

• At a turning point, velocity is zero and the object reverses direction.

Integrals in Kinematics

Integrals are used to find the total change over an interval:

• Displacement from Velocity: $\Delta x = \int_{t_i}^{t_f} v\,dt$

• The area under the velocity-time graph between two times gives the displacement.

Uniform Motion and Uniform Acceleration

Uniform Motion

Uniform motion occurs when an object moves at constant velocity:

• Equation: $x = x_0 + v t$

• Position changes linearly with time.

Uniform Acceleration

Uniform acceleration means the velocity changes at a constant rate:

• Key Equations:

  • $v_f = v_i + a t$

  • $x_f = x_i + v_i t + \frac{1}{2} a t^2$

  • $v_f^2 = v_i^2 + 2 a (x_f - x_i)$

• These equations are valid only when acceleration is constant.

Free Fall and Gravity

Free Fall

Free fall is the motion of an object under the influence of gravity alone:

• Acceleration due to Gravity: $g = 9.80\ \text{m/s}^2$ (downward, near Earth's surface)

• All objects in free fall experience the same acceleration, regardless of mass.

• Example: Dropping a rock from a height of 20 m, use $v_f^2 = v_i^2 + 2 a (x_f - x_i)$ to find the impact velocity.

Inclined Planes and Simple Machines

Inclined Plane

An inclined plane is a simple machine that reduces the force needed to lift an object by increasing the distance over which the force is applied.

• Acceleration on Incline: $a = g \sin \theta$ (where $\theta$ is the angle of incline)

• The component of gravity along the ramp is less than the full gravitational acceleration.

• Historical Note: Inclined planes have been used since ancient times for construction and transport.

Summary Table: Key Kinematic Quantities

Quantity	Definition	SI Unit	Formula
Position (x)	Location along a line	meter (m)	–
Displacement (Δx)	Change in position	meter (m)	$\Delta x = x_f - x_i$
Velocity (v)	Rate of change of position	m/s	$v = \frac{dx}{dt}$
Acceleration (a)	Rate of change of velocity	m/s2	$a = \frac{dv}{dt}$
Average Speed	Total distance / total time	m/s	$\text{Average speed} = \frac{\text{distance}}{\text{time}}$
Average Velocity	Total displacement / total time	m/s	$\text{Average velocity} = \frac{\Delta x}{\Delta t}$

Additional info:

• Students are not required to perform calculus (differentiation or integration) on exams, but should understand the conceptual meaning of derivatives (slopes) and integrals (areas under curves) in kinematics.

• Simple machines, such as the inclined plane, are introduced as applications of kinematics and force analysis.