Motion in One Dimension: Physics with Calculus Study Notes

Motion in One Dimension

Introduction to Linear Motion

Motion in one dimension involves the study of objects moving along a straight line, either horizontally or vertically. This foundational topic introduces key concepts such as position, velocity, acceleration, and the mathematical relationships that describe motion.

Describing Motion

Position and Coordinate Systems

To analyze motion, we use a coordinate system. The x-axis is typically used for horizontal motion, while the y-axis is used for vertical motion. The position of an object is its location relative to an origin.

• Position (x or y): The location of an object along a straight line, measured from a chosen origin.

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

Motion Diagrams

Motion diagrams represent an object's position at successive time intervals. They help visualize whether an object is moving at constant speed, speeding up, or slowing down.

Position vs. Time Graphs

A position-versus-time graph plots an object's position on the vertical axis and time on the horizontal axis. The slope of this graph indicates the object's velocity.

• Steeper slope: Faster speed.

• Positive slope: Motion in the positive direction.

• Negative slope: Motion in the negative direction.

Velocity

Velocity is the rate of change of position with respect to time. It is a vector quantity, meaning it has both magnitude and direction.

• Average velocity (vavg): $v_{avg} = \frac{\Delta x}{\Delta t}$

• Instantaneous velocity: The velocity at a specific instant, given by the slope of the tangent to the position-time graph at that point.

Velocity vs. Time Graphs

A velocity-versus-time graph shows how velocity changes over time. The area under the curve represents the object's displacement.

• Constant velocity: Horizontal line.

• Changing velocity: Sloped or curved line.

From Position to Velocity and Vice Versa

The velocity at any instant is the slope of the position-time graph. Conversely, the position can be found by calculating the area under the velocity-time graph.

Uniform Motion

Definition and Representation

Uniform motion refers to straight-line motion with constant velocity. Equal displacements occur during equal time intervals, and the position-time graph is a straight line.

• Equation for uniform motion: $x = x_0 + v_x t$

Proportional Relationships

In uniform motion, displacement is proportional to time. If you double the time, the displacement also doubles. This is an example of a proportional relationship, $y = Cx$.

Instantaneous Velocity

Definition and Calculation

Instantaneous velocity is the velocity of an object at a specific instant. It is found by taking the slope of the tangent to the position-time curve at that point.

• Mathematical definition: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

Displacement from Velocity-Time Graphs

The area under a velocity-time graph, even for non-uniform motion, gives the displacement.

Acceleration

Definition and Units

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity and can be positive or negative depending on the direction of velocity change.

• Average acceleration (a): $a = \frac{\Delta v}{\Delta t}$

• SI unit: meters per second squared (m/s2)

Sign of Acceleration

The sign of acceleration depends on both the direction of motion and whether the object is speeding up or slowing down. If velocity and acceleration have the same sign, the object speeds up; if they have opposite signs, it slows down.

Motion with Constant Acceleration

Kinematic Equations

When acceleration is constant, the following equations describe the motion:

• $v_x = v_{x0} + a_x t$

• $x = x_0 + v_{x0} t + \frac{1}{2} a_x t^2$

• $v_x^2 = v_{x0}^2 + 2 a_x (x - x_0)$

Quadratic Relationships

For constant acceleration, position is a quadratic function of time. The graph of position versus time is a parabola.

• Quadratic relationship: $y = Cx^2$

Free Fall

Definition and Properties

Free fall is the motion of an object under the influence of gravity alone, with no other forces acting. All objects in free fall near Earth's surface experience the same acceleration, regardless of mass.

• Free-fall acceleration (g): $g = 9.8\ \mathrm{m/s^2}$ (always directed downward)

• Use kinematic equations with $a = -g$ for upward motion and $a = g$ for downward motion.

Problem-Solving Strategies

Four-Step Approach

1. Strategize: Identify the type of problem and relevant principles.

2. Prepare: Draw diagrams, define variables, and list knowns and unknowns.

3. Solve: Apply appropriate equations and perform calculations.

4. Assess: Check units, reasonableness, and completeness of the answer.

Visual Overview

A comprehensive visual overview may include a motion diagram, pictorial representation, list of values, and graphical representation. These tools help organize information and clarify the solution process.

Summary of Key Concepts

• Velocity is the rate of change of position: $v = \frac{dx}{dt}$

• Acceleration is the rate of change of velocity: $a = \frac{dv}{dt}$

• Uniform motion: Constant velocity, straight-line position-time graph.

• Constant acceleration: Linear velocity-time graph, parabolic position-time graph.

• Free fall: Special case of constant acceleration with $a = -g$.