Position, velocity, and acceleration
Read Tipler and Mosca (2004) chapter 2 (Motion in
1D); Pelcovits and Farkas
(2024) chapter 2 (Kinematics)
Kinematics
Kinematics is the quantitative study of motion. To describe how an object moves, we typically describe its position, velocity, and acceleration in some useful coordinate system / coordinate frame.
Position, velocity, acceleration
Position (and displacement)
Position is a vectorA vector is a number and a direction. To convince
yourself direction matters, imagine flying up from the ground \(\qty{33}{\meter}\), versus flying down into
the ground \(\qty{33}{\meter}\); are
those different? Would the direction matter? Yes!
and describes where an object is in space relative to an
established coordinate system. Typical SI units for position are m. I
normally use \(\vec{r}\), \(x\), \(y\), or \(z\) as variables to describe position,
often with an arrow over them to remind myself they are vectors.
To measure position, we would need to have some sort of coordinate system with an agreed upon origin and axes.
Displacement is the position relative to some initial position. If I was using \(x\) to measure position, the displacement would be \(x-x_0\), where \(x_0\) represents the initial position. Displacement in general is also a vector, e.g. \(\vec{r}-\vec{r}_0\).
Velocity
Velocity is also a vector and describes the time rate of change of position. Its units are \(\unit{\meter\per\second}\). I normally use \(\vec{v}\) to represent velocity. Considering \(\Delta\) or \(d\) as a “change in”, velocity becomes \[\begin{aligned} \text{velocity}, [\unit{\meter\per\second}] &= \dfrac{\text{change in position}}{\text{change in time}} \\ &= \dfrac{\Delta \vec{x}}{\Delta t} \\ &= \dfrac{d\vec{x}}{dt} \end{aligned}\]
The last form tells us that velocity is the (time) derivative of position. The “time rate of change of position” relationship means that velocity is like the slope of a position versus time graph. Later, when you cover anti-derivatives and integrals in calculus, you will see that, going the other way, position is like the area under a velocity versus time graph.
Velocity feels like going fast. The speedometer in your car indicates (in a way) your velocity; in actuality, it indicates the magnitude of your velocity.
Acceleration
Acceleration is obtained from doing a Madlibs sort of thing with our pattern of \(\dfrac{\Delta\text{something}}{\Delta t}\). We take it as the time rate of change of velocity. Its units are \(\unit{\meter\per\second\squared}\). I normally use \(\vec{a}\) to represent acceleration. \[\begin{aligned} \text{acceleration}, [\unit{\meter\per\second\squared}] &= \dfrac{\text{change in velociy}}{\text{change in time}} \\ &= \dfrac{\Delta \vec{v}}{\Delta t} \\ &= \dfrac{d \vec{v}}{dt} \end{aligned}\]
The “time rate of change of velocity” relationship means that
acceleration is like the slopeRemember slope as rise over run in \(y=mx+b\) in your math classes; here compare
to \(x=vt+x_0\) and \(v=at+v_0\).
of a velocity versus time graph,and that
velocity is like the area under an acceleration versus time
graph.
Acceleration is the (time) derivative of velocity, or the second derivative of position. \[\vec{a} = \dfrac{d\vec{v}}{dt} = \dfrac{d^2\vec{x}}{dt^2}\]
Acceleration is what you feelStudents sometimes confuse velocity and acceleration
because in regular English language usage they are used similarly. In
physics we must be more precise.
when you are in a Lamborghini at a stop light waiting to
get donuts, the light turns green, and your floor it. Acceleration pulls
you back into your seat. It is the g’s felt by a fighter pilot. It is
what you pay for when you ride a roller coaster at Great Adventures and
vomit. It is also what you feel when you see a kitten and slam on the
brakes to not hit it. It is not going fast, rather it is what you feel
when your speed is changing to go faster or slower.
Calculus relationships between position, velocity, and acceleration
The three quantities we are discussing are related by
derivativesWe might formally define the derivative of a function
\(f(x)\) as \(\dfrac{df}{dx} = \lim_{\Delta x\to 0}
\dfrac{f(x+\Delta x)-f(x)}{\Delta x}\). Sometimes as shorthand we
write this using prime notation, \(f'(x)=\dfrac{df}{dx}\). When dealing
specifically with the time derivative, we might also write this
shorthand using dots, \(\dfrac{dx}{dt} =
\dot{x} = v\).
; position and its first (velocity) and second
(acceleration) derivative. Derivatives were invented in part to deal
with such quantities.
\[\begin{aligned} \vec{x}, \text{position} &= \vec{x} \\ \vec{v}, \text{velocity} &= \dfrac{d\vec{x}}{dt} \\ \vec{a}, \text{acceleration} &= \dfrac{d\vec{v}}{dt} = \dfrac{d^2\vec{x}}{dt^2} \end{aligned}\]
What if you have acceleration and wanted velocity or position? Is
there a function that is like the inverse of the derivative? The answer
is yes, there are anti-derivativesYou will get to integrals in maybe a few months in
calculus class
and they occur when taking the integral
of a function. The integral is like taking the area under the curve and
is denoted with the funny stylized S (for sum) symbol \(\int\), which means you sum up little
slices of the function that are \(f(t)\) high and \(dt\) wide, while letting the \(dt\) become very smallFor this unit, other than just introducing the idea of
areas under a curve, I will try not to use integrals as it
would be too far in advance of your calculus class.
.
\[\begin{aligned}
\vec{a}, \text{acceleration} &= \vec{a} \\
\vec{v}, \text{velocity} &= \int \vec{a}\, dt \\
\vec{x}, \text{position} &= \int \vec{v}\, dt
\end{aligned}\] The third derivative of position, \(\dfrac{d^3\vec{x}}{dt^3}\) is called the
jerk. The trajectories of voluntary arm movements in monkeys and humans
have been shown to minimize the jerk. Beyond jerk, the fourth, fifth,
and sixth derivatives of position are the snap, crackle, and pop. I know
of no valid published scientific use of these, but the names are cool
and rice krispies are tasty.
Testing figures
This caption should appear in the margins.
This should appear as a margin figure from The
Mechanical Universe. More caption information.
.
See also
The Mechanical Universe, several episodes:
https://www.youtube.com/watch?v=w6ynSbzPhjc&list=PL8_xPU5epJddRABXqJ5h5G0dk-XGtA5cZ&index=7
https://www.youtube.com/watch?v=f7MTOb8GUwk&list=PL8_xPU5epJddRABXqJ5h5G0dk-XGtA5cZ&index=8
Also check out:
Summary
The relationships between position, velocity, and acceleration are not given on your AP Physics C Mechanics equation sheet. They are fundamental to the class and you are expected to just know these as if they are second nature.