Motion in One Dimension

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>

Changing Acceleration

Particles may also change their acceleration over time. Perhaps it represents a car that accelerates and brakes, accelerates and brakes.

Bouncing Ball

Particles may suddenly change their velocity as well, such as when they bounce off the ground or a wall.

Etc...

You can probably come up with other possible equations for position, velocity, and acceleration.

Average Velocity

Given a particle is at position <math>x_0</math> at time <math>t_0</math> and then at position <math>x_1</math> at time <math>t_1</math>, we can calculate its average velocity between those two times. It is simply:\

<math>v_{avg} = {x_1 - x_0 \over t_1 - t_0} = {\Delta x \over \Delta t}</math>

Instantaneous velocity

Knowing the average velocity gives only a part of the picture. We want to know the velocity at any given moment in time. We can do so using elementary calculus.

<math>v = \lim_{\Delta t \rightarrow 0} {\Delta x \over \Delta t} = {dx \over dt} = x'</math>

Accelerated Motion

We can calculate the average velocity by looking at how the velocities change over time. We can also find the instantaneous velocity by finding the derivative of the velocity.

<math>a = \lim_{\Delta t \rightarrow 0} {\Delta v \over \Delta t} = {dv \over dt} = v' = {d^2 x \over dt^2} = x</math>

Source

• Chapter 2 of HRK