Motion in One Dimension
Contents
Why One Dimension?
Understanding how things move in one dimension will help us understand many dimensions. That's because in one dimension, position can be described by a single number, as opposed to multiple dimensions where vectors are required.
Why a particle?
By looking only at things that have no size, AKA, particles, we eliminate many complexities that arise from motions such as spinning. This also simplifies the math. It prepares us to analyze objects that have dimensions because we can treat each part of the larger object as a particle.
As long as objects in real life behave like particles, that is, they don't spin or change shape, we should be able to approximate their motion with a single particle of similar mass.
Describing Motion with Math
We use a function that takes time as a parameter to describe the position <math>x(t)</math>, velocity <math>v(t)</math>, and acceleration <math>a(t)</math> of particles. I'll demonstrate this below.
No Motion
First, we consider particles that do not move at all. Their position never changes. The velocity and acceleration are always zero.
<math> \begin{align}
x(t) &= x_0 \\ v(t) &= 0 \\ a(t) &= 0 \\ \end{align} </math>
Constant Velocity
Next, let's consider a particle moving at constant velocity. Its acceleration is always zero, and its velocity is constant. The position is simply the initial position plus the velocity times the time.
<math> \begin{align} x(t) &= x_0 + v_0 t \\ v(t) &= v_0 \\ a(t) &= 0 \\ \end{align} </math>
Constant Acceleration
Particles that are constantly accelerating have a velocity that changes and a position that also varies.
<math> \begin{align} x(t) &= x_0 + v_0 t + 1/2 a_0 t^2 \\ v(t) &= v_0 + a_0 t \\ a(t) &= a_0 \\ \end{align} </math>
Source
- Chapter 2 of HRK
