Module1 Motion in 1D
Defining Physics
The dictionary defines physics to be “a science that deals with matter and energy and their interactions”. I would like to expand upon this overly broad definition with a list of characteristics:
Physics is:
- Based on observations and measurements
- Used to identify fundamental laws of nature
- Focused on the minute details of the physical world
- A very mature field, capable explaining incredibly complex systems
- Not sophisticated enough to tackle problems in biology, and many problems in chemistry
Where does Physics fit?
I would like to start out with a brief overview of how physics fits into the scientific world. Physics was the first modern science to be studied and understood by humans. The very basics of the field were studied by the ancient Greeks and Romans, but major advancements in the field started happening during the enlightenment (1700s). This is when philosophers like Sir Isaac Newton began to use sophisticated mathematics to model our universe.
For the last three hundred years, brilliant mathematicians and scientists have advanced the field to the point where we can predict the dynamics of large, exotic objects like neutron stars and tiny systems like a pair of coherent electrons, and all sorts of objects in between. But physics cannot solve every problem in our natural world, in fact it struggles to give any insight into the dynamics of even simple biological systems.
Scales of size
The Sciences have been developed over the last few hundred years to solve specific types of problems. Each science must rely on a set of assumptions when trying to understand physical system. Physics, being the most rigid and fundamental of the Sciences, tries to only use the base level physical laws to build up complicated theories, like quantum mechanics and relativity. This creates a large problem when physicists try to solve a complicated problem like a protein folding or a plant cell photosynthesizing light. A higher level science like biology, will make assumptions about the cells which are not based upon foundational physics laws, instead they rely on experimental observation. These generalizations help solve the complicated problem, but sacrifice fundamental understanding in the process. Both types of science have their Merit, and each type is suited to specific types of phenomena. Physics has no chance of solving a complicated biological problem, it is simply too hard to mathematically model.
Below I have included a chart which sorts the size of object from biggest to smallest. Take a look and try to think about where your field of study fits into this chart.
| Biggest Objects | Universe, Galaxies, Stars | PHYSICS! Astrophysics, Relativity, Cosmology |
| Planets | Geology | |
| Human Scale | Physics, Chem, Bio, Sociology, Psycology, etc… | |
| Proteins, Polymers | Biology | |
| Molecules | Chemistry | |
| Atoms | Physics and Chemistry | |
| Smallest Objects | Quarks, Sub-Atomic Particles | PHYSICS! Quantum physics, Nuclear physics |
Physics is very good at describing the smallest of objects in our world. This is because simple objects have simple mathematical models, and those models can be studied and integrated into the field. Physics also has showing great promise describing the largest objects in our universe. The reason physics still applies to these objects is they are mostly uniform and again simple models can be created to describe the phenomena.
Right in the middle of this scale is the scale of a human. This scale includes things like books in computers and houses and it is this size object that we will be focusing on in this course. Physicists like to call the study of objects of this size Mechanics.
Units Of Measurement
In order to have a discussion about physics (or science in general), we first need to agree upon a common set of physical measurements. For the field of mechanics, there are three fundamental quantities which we will need to define. All other units of measurement will be defined from these three. As you can imagine, these units must be defined as rigidly as possible so that the foundation of the field is structurally sound. The definitions have been chosen to be agnostic of the fact that we are humans and we are living on planet Earth. These definitions could be transmitted to an alien world and a scientifically advanced alien race would still be able to determine what our standard of measurement is.
Scientists use a specific set of units known as SI units (Links to an external site.) (standard international). These units are better known as metric. This course will not use any imperial units such as pounds or inches.
Length
The fundamental unit of length in physics is the meter. The meter is defined to be the distance that light travels in a vacuum in 1/299,792,458 seconds. Indeed, this is a very rigid definition!
We will use meters for every distance measurement in this course. In American terms, a meter is slightly larger than a yard, a human is approximately 2 meters tall, and a football field is approximately 90 meters long.
1 meter = 3.28 feet = 1.09 yards
Mass
The fundamental unit for mass is the kilogram. This Mass used to be defined using a block of metal kept in a vault somewhere in Paris. This is a very poor way to define a unit! In fact in 2019, the SI definition for mass has been changed to be derived from the Planck constant.
Note: Please be sure not to confuse Mass with weight (as you will see later in this course, a weight it is actually a force).
We will use Kilogram for every mass measurement in this course. In American terms, where kilogram is 2.2 lb, an average human has a mass of around 60kg to 70 kg.
1 kg = 2.2 lbs
Time
The fundamental unit for time is the second. This is a very familiar unit to everyone. It has a similarly rigid definition: 9,192,631,770 times the period of oscillation of a Cesium-133 atom.
We will use seconds for every time measurement in this course.
Conversion of units
Conveniently, units obey normal algebraic and mathematical manipulation. This fact allows us to convert between different units using a simple mathematical prescription.
Common SI Units
| Quantity | Variable | Unit |
|---|---|---|
| Position | x | [m] |
| Area | A | [m2] |
| Velocity | v | [m/s] |
| Acceleration | a | [m/s2] |
| Time | t | [s] |
| Mass | m | [kg] |
| Force | F | [N] = [kg*m/s2] |
| Energy | E | [J] = [kg*m2/s2] |
Displacement:
The position of an object is described using the variable \(x\). The first quantity to study is related to the position of the object, it is known as the displacement \(\Delta x\). The displacement is the difference between the positions:
\[\Delta x = x_f - x_i\]
Note: the distance is not necessarily equal to the displacement. The distance accounts for the entire path length the object has traveled, whereas the displacement only measures the change imposition. The position will not be used in our physics equations, so instead please always think of the displacement when calculating positional change.
Velocity:
The next quantity, known as the velocity, describes the rate of change of position. This is a fancy mathematical way to say “how fast are you moving?”.
\[v = \frac{\Delta x}{\Delta t}\]
I like to read this equation in English as the following phrase: “the velocity changes the position”. if you have a non-zero velocity, your position must change! please try to think about driving in a car. if your car has a velocity (given in miles per hour), this means it’s moving or changing position. As long as the car has a velocity, it is required to move.
Acceleration:
the final quantity to study in this chapter is the acceleration. it is very easy to misunderstand what the acceleration does. the acceleration is the rate of change of the velocity.
\[a = \frac{\Delta v}{\Delta t}\]
I like to read this equation in English as: “the acceleration changes the velocity”. If you have a non-zero value for acceleration, your velocity is required to change continuously. Again try to imagine yourself in a car. If you hit the accelerator and speed the car up, you have an acceleration, which continually changes the velocity. You could also press the brake, which will change your velocity in the opposite (possibly negative) direction. Both the accelerator and the brake are changing your velocity and therefore cause an acceleration.
Free Fall
Using Calculus to derive Kinematics
The so called “kinematic equations” are the most used equations when solving free fall and ballistic physics problems. They are calculated using some basic definitions of the position, velocity and acceleration:
\[v = \frac{dx}{dt} \qquad \qquad a = \frac{dv}{dt}\]
These definitions for the velocity and acceleration were discussed previously, but the equations of motion still must be computed. In order to calculate the kinematic equations, you will need to make an assumption about the motion. Some quantity will need to be held constant (value cannot change). Below I will discuss 4 cases:
Constant Position:
This may be the most boring of all of the cases. If the position is constant, no movement will occur (may be obvious). The way to identify this with the mathematics is to look at the definition of the velocity: \(v(t) = \frac{dx(t)}{dt} = \frac{d}{dt} x(t)= 0\)
(The final answer can be read in English: the derivative of a constant is zero.) The kinematic equation for this is extremely boring:
\[ x_f = x_i\]
Constant Velocity: Here we get into something more interesting. If the velocity is constant, the position will change, but the acceleration will not change (\(\Delta v =0\)). You can integrate the velocity equation to find the kinematic equation:
\[ x_f = x_i + v\, t\]
Constant Acceleration:
Objects moving under constant acceleration is a very common physics problem for this topic. This is because the acceleration due to gravity is approximately constant for objects on the surface of the earth. We are all confined to live on the surface of the earth, and therefore are extremely used to this acceleration constantly pulling us down. The gravitational acceleration is used so frequently that it gets its own symbol in physics: \(g = 9.8\, m/s^2\). As you will see in the below video, the kinematic equations are normally derived using the assumption of “constant acceleration”. I will now write down the kinematic equations which you will frequently use in this course. If you want to see a derivation, please see the below video:
\[x_f = x_i + v_i \, t + \tfrac12 a t^2\] \[v_f = v_i + a\, t\] \[v_f^2 = v_i^2 + 2a (x_f - x_i)\]
Constant Jerk:
I would also like to talk about the extension of this topic to situations where the acceleration is not constant. Instead lets assume that the acceleration changes linearly in time. The rate that the acceleration changes (slope) is known as the Jerk (Links to an external site.) (units of m/s3). If the Jerk is constant then the equation of motion will need one more integration to arrive at the final result for the position as a function of time. I hope you can see the pattern of increasing level of polynomials from the above cases:
\[x_f = x_i + v_i \, t + \tfrac12 a_i\, t^2 + \tfrac16 J\, t^3\]
Examples
I have some solutions for 1D Kinematics problems if you want to take a look at how I approach these problems. I find my steps to be fairly close to the steps outlined in the book in the example problems:
Please note the numbers in the example problems are not the same as the numbers in your book…
Mathematics Review - Polynomial Equations
Solving kinematics problems requires you know all of the mathematical tools for soltion of linear and quadratic equaitons.
Polynomial Equations
A quadratic equation is any polynomial equaiton with maximum power of 2. The generic form is given as:
\[f(t) = a\,t^2 + b\,t + c\]
where t is the variable and a,b,c are constants. This function can have 3 different shapes:
| Shape | criteria | Funciton |
|---|---|---|
| Constant | \(a\) and \(b\) are both zero | \(f(t) = c\) |
| Linear | \(a\) is zero and \(b\) is non-zero | \(f(t) = b\,t + c\) |
| Parabolic | \(a\) must be non-zero | \(f(t) = a\,t^2 + b\,t + c\) |
This type of function is used extensively in kinematics (first few chapters of the course). The path resembles the trajectory of an object thrown in earth’s gravitational field. Note that the curverature (upward or downward) is determined by the sign of \(a\).
The solution to kinematics problems will involve all three of these types of curves, Please take your time to understand them. Next, you will inspect the position velocity and acceleration of a fairly generic 1D kinematics problem.
Example Problem
QUESTION: A student stands at the top of a 12.5m tall building and throws a ball upwards with an initial velocity of 10m/s. Plot the position velocity and acceleration of the ball as it travels through the air.
Recall that the Kinematic equaitons look very similar to the quadratic or linear equations:
\[a(t) = a\] \[v(t) = v_i + a\,t\] \[x(t) = x_i + v_i\,t + \tfrac{1}{2} a \, t^2\]
where for this example, the initial conditions are:
- \(x_i = 12.5~m\)
- \(v_i = 10~m/s\)
- \(a = -9.8~m/s^2\)
Example Problem: Acceleration
\[a(t) = \text{constant}\]
This function is very boring. The function evaluates to the constant no matter what the input is. If you make a figure of this function, it would just look like a flat line (only one constant value on the vertical axis, no matter what you chose for the x-axis).
Recall the acceleration is required to be constant when using the standard kinematics equations.
Example Problem: Velocity
\[v(t) = v_i + a\, t\] where m is the slope of the line and b is the y-intercept. This function has two constants:
- Slope: the steepness of the line (rise over run). If the slope is positive, the line moves up and to the right. If slope is negative, the line moves down and to the right.
- y-intercept: The location where the line crosses the vertical axis.
<>:13: SyntaxWarning: invalid escape sequence '\,'
<>:13: SyntaxWarning: invalid escape sequence '\,'
/tmp/ipykernel_24614/3761710615.py:13: SyntaxWarning: invalid escape sequence '\,'
plt.title("Velocity: $v(t) = v_i + a \, t$")
Example Problem: Position
The position under constant acceleration takes the form of a parabola. The parabola will curve downward if the quadratic term (\(a\)) is negative.
\[x(t) = x_i + v_i \, t + \tfrac{1}{2} a\, t^2\]
<>:17: SyntaxWarning: invalid escape sequence '\,'
<>:17: SyntaxWarning: invalid escape sequence '\,'
/tmp/ipykernel_24614/1166577912.py:17: SyntaxWarning: invalid escape sequence '\,'
plt.title("Position: $x(t) = x_i + v_i \, t + a \, t^2$")
Kinematics: Motion under constant acceleration
Now you can manipulate each of the inital conditions to see what happens to the position, velocity and acceleration of an object. The initial conditions for the simulation are set to be \(a = -9.8\,m/s^2\), \(v_i = 10 \, m/s\) and \(x_i = 12.5\,m\).
Please take a moment to answer these questions:
- Does the acceleration change at the top of the trajectory? Does the acceleration change at any point in the entire motion? (Please remember this is required for the kinematics equations).
- Manipulate the sliders such that the velocity a constant value (horizontal line)? What does the acceleration need to be to have constant velocity?
- Manipulate the sliders such that the position is constant. What does the accleration and the velocity need to be for the motion to have constant position?
- What changes about the velocity and position if the acceleration is changed to be positive? Which direction does the parabola open (curverature)?