Motion in One Dimension

Understanding the basics of linear motion, velocity, and displacement

Introduction to Motion in One Dimension

Motion in one dimension, also known as linear motion, describes the movement of objects along a straight line. This is the simplest form of motion and serves as the foundation for understanding more complex movements. We'll explore how to describe position, calculate displacement and velocity, and understand the relationship between these fundamental quantities.

In this section:
  1. Position and Displacement
  2. Velocity and Speed
  3. Acceleration in One Dimension
  4. Kinematic Equations
  5. Free Fall Motion

Position and Displacement

The position of an object refers to its location relative to a chosen reference point, often denoted as the origin (\(x = 0\)). Position is a vector quantity in higher dimensions, but in one dimension, it's simply a coordinate along a line.

Displacement is the change in position of an object, representing both the distance and direction from the initial to the final position. Unlike distance, displacement can be negative, indicating direction.

Displacement Formula
\[\Delta x = x_f - x_i\]
Where:
  • \(\Delta x\) = displacement (in meters, m)
  • \(x_f\) = final position (in meters, m)
  • \(x_i\) = initial position (in meters, m)
Example 1:

If a car moves from \(x = 5\) m to \(x = 12\) m, the displacement is:

\[\Delta x = 12 \text{ m} - 5 \text{ m} = 7 \text{ m}\]

The positive value indicates the car moved in the positive direction.

Velocity and Speed

Velocity is the rate of change of displacement with respect to time. It is a vector quantity that includes both magnitude (speed) and direction. Speed is the magnitude of velocity and is always positive.

Average Velocity Formula
\[v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}\]
Where:
  • \(v_{avg}\) = average velocity (in meters per second, m/s)
  • \(\Delta x\) = displacement (in meters, m)
  • \(\Delta t\) = time interval (in seconds, s)

Instantaneous velocity is the velocity at a specific moment in time, defined as the limit of the average velocity as the time interval approaches zero:

Instantaneous Velocity
\[v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}\]

Acceleration in One Dimension

Acceleration is the rate of change of velocity with respect to time. It describes how quickly an object's velocity changes and can be positive (speeding up in the positive direction) or negative (slowing down or speeding up in the negative direction).

Average Acceleration Formula
\[a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}\]
Where:
  • \(a_{avg}\) = average acceleration (in m/s²)
  • \(\Delta v\) = change in velocity (in m/s)
  • \(\Delta t\) = time interval (in seconds, s)
  • \(v_f\) = final velocity, \(v_i\) = initial velocity

Kinematic Equations

For motion with constant acceleration, we can use the following kinematic equations. These are the fundamental tools for solving one-dimensional motion problems:

Kinematic Equations for Constant Acceleration
\[1.\quad v = v_0 + at\] \[2.\quad x = x_0 + v_0 t + \frac{1}{2}at^2\] \[3.\quad v^2 = v_0^2 + 2a(x - x_0)\] \[4.\quad x - x_0 = \frac{1}{2}(v_0 + v)t\]
Where:
  • \(x\) = final position, \(x_0\) = initial position
  • \(v\) = final velocity, \(v_0\) = initial velocity
  • \(a\) = constant acceleration
  • \(t\) = time elapsed

Free Fall Motion

When an object falls under the influence of gravity alone (ignoring air resistance), it experiences free fall. Near the Earth's surface, all objects fall with the same acceleration due to gravity, regardless of their mass.

Acceleration Due to Gravity
\[g = 9.8 \text{ m/s}^2\]

Note: This value is directed downward toward Earth's center.

For free fall motion, we use the same kinematic equations with \(a = -g\) (if we choose upward as positive) or \(a = +g\) (if we choose downward as positive).

Example 2: Free Fall Problem

A ball is dropped from a height of 20 meters. How long does it take to reach the ground, and what is its velocity just before impact?

Step 1: Find the time using \(x = x_0 + v_0 t + \frac{1}{2}at^2\)

Taking downward as positive: \(x_0 = 0\), \(x = 20\) m, \(v_0 = 0\), \(a = g = 9.8\) m/s²

\[20 = 0 + 0 \cdot t + \frac{1}{2}(9.8)t^2\]

\[t^2 = \frac{40}{9.8} = 4.08\]

\[t = 2.02 \text{ seconds}\]

Step 2: Find velocity using \(v = v_0 + at\)

\[v = 0 + 9.8 \times 2.02 = 19.8 \text{ m/s downward}\]

Key Concepts Summary

  • Position is location relative to a reference point
  • Displacement is change in position (can be negative)
  • Velocity is the rate of change of displacement
  • Acceleration is the rate of change of velocity
  • Kinematic equations apply only when acceleration is constant
  • Free fall is motion under gravity alone with \(a = g = 9.8\) m/s²
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