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>

Understanding Acceleration

At this point, we need to work at distinguishing position, velocity, and acceleration. Their cousins in the English language only confuse matters, so we must be precise in our words.

• Position is where the particle is at any given point in time. Just because something has non-zero velocity or acceleration does not mean that the position at a particular point in time is not zero. Likewise, a zero velocity or acceleration does not imply anything about where the particle is.
• Velocity describes how fast the position is changing (what we call speed), and in what direction. A particle can be at position zero and have a non-zero velocity.
• Acceleration describes how fast the velocity is changing, and in what direction. There isn't a good word in plain English to describe this concept.

Little children inherently understand that going fast (a high velocity) doesn't mean you're going fast (a high acceleration.) They don't have the words to describe this, but they know all about it.

For instance, suppose I jam on the accelerator in a sports car from a stop sign. My velocity initially is 0, and indeed, my speedometer agrees with that. However, the child in the back seat feels the acceleration, and they know in a few moments I will have a high velocity. On the other hand, they feel that hurling down the freeway at 60 MPH with no acceleration (since I am neither going faster or slower) feels much the same as standing still at a stoplight or cruising down the city streets at 35 MPH, except for there is a bit more bumps.

It is at this point where you no longer have any excuse to confuse acceleration and velocity. Keep the distinction in your mind.

Constant Acceleration

It is useful to look at the math behind simple constant acceleration. That is, <math>a(t) = a_0 \ne 0</math>. The velocity can be derived using simple differential equations:

<math> \begin{align} a(t) &= a_0 \ne 0 \\ {dv \over dt} &= a(t) = a_0 \\ dv &= a_0 dt \\ \int dv &= \int a_0 dt \\ v + C_v &= a_0 t + C_t \\ v(t) &= v_0 + a_0 t \\ \end{align} </math>

Likewise, we can do the same to derive the position.

<math> \begin{align} v(t) &= v_0 + a_0 t \\ {dx \over dt} &= v(t) \\ {dx \over dt} &= v_0 + a_0 t\\ dx &= (v_0 + a_0 t) dt \\ \int dx &= \int v_0 + a_0 t dt \\ x + C_x &= v_0 t + 1/2 a_0 t^2 + C_v \\ x &= x_0 + v_0 t + 1/2 a_0 t^2 \\ \end{align} </math>

Free Fall

Near the earth's surface, objects tend to accelerate uniformly towards the ground. This was a discovery of Galileo, long before Newton. The acceleration due to gravity by experiment is <math>g \approx 9.81 m/s^2</math>.

If we deal only with 1 dimensional motion -- up and down -- then we can calculate the position of any object at any given time given the initial velocity <math>v_0</math> and the initial position <math>x_0</math>.

Of course, we are approximating a great deal here, including factors such as wind resistance and other minute but measurable effects. However, as long as we worry only about the most significant digits, these discrepancies tend to be inobservable, particularly when we are using imprecise equipment such as movie cameras and measuring tapes. So the basic equations about are simply good enough.

What Galileo Did

It is at this time that we can abandon the school-child's understanding of Galileo's experiments with gravity. Unfortunately, what you have probably learned about him and his experiment is wrong in important but subtle ways.

Aristotle proposed that heavier things fall faster, which, if you consider terminal velocity only, tends to be true (as we shall see later.) A simple demonstration of this is dropping a ball and a piece of paper. The paper floats to the ground, while the ball quickly achieves its destination. However, if you crumple up the same piece of paper, you'll see that the paper and ball fall at about the same speed. This shows that the shape of the object has much more to do with the speed of falling than the mass, and completely discredits Aristotle.

(Side note: A lot of modern physics is built upon investigating claims by Aristotle, and finding them simply not true. You should continue this tradition by investigating all the claims I make, or any Physics teacher, and demand experimental proof before accepting anything as true.)

Galileo did not have a way to eliminate air resistance. The story of him dropping two objects from the Tower of Pisa is almost certainly a legend. Instead, Galileo rolled balls of similar shape down an incline. Since the objects descended much more slowly, they were easier to measure. Of significance, air resistance for slow objects is almost negligible. He found that the distances covered in successive time periods were the sums of the odd numbers, which we simply know as the squares. This showed Galileo that the speed increased at a constant rate for all objects, and gave him the conclusion that the acceleration due to gravity is a universal constant (for everything near the earth's surface, of course.) The mass of the object had nothing to do with it.

Measuring g

Measuring g accurately can be done with increasingly complex and precise machines. We know today that g varies by measurable amounts between the floors of buildings, and even based on what is immediately below the surface of the earth. Despite these accurate measurements, no one has detected any dependence on the mass of the falling object. The gravitational mass and momentum mass seem to be exactly equal to one another. (More on what that means later.)

Source

• Chapter 2 of HRK