Kinematics and Calculus
Discussion
motion with constant acceleration
Calculus is an advanced math topic, but it makes deriving two of the three equations of motion much simpler. By definition, acceleration is the first derivative of velocity with respect to time. Take the operation in that definition and reverse it. Instead of differentiating velocity to find acceleration, integrate acceleration to find velocity. This gives us the velocity-time equation. If we assume acceleration is constant, we get the so-called first equation of motion [1].
| a | = |
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| dv | = | a dt | |||||||||||||
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= |
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| v − v0 | = | at | |||||||||||||
| v | = | v0 + at [1] | |||||||||||||
Again by definition, velocity is the first derivative of position with respect to time. Reverse this operation. Instead of differentiating position to find velocity, integrate velocity to find position. This gives us the position-time equation for constant acceleration, also known as the second equation of motion [2].
| v | = |
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| ds | = | v dt | ||||||||||||||
| ds | = | (v0 + at) dt | ||||||||||||||
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= |
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| s − s0 | = | v0t + ½at2 | ||||||||||||||
| s | = | s0 + v0t + ½at2 [2] | ||||||||||||||
Unlike the first and second equations of motion, there is no obvious way to derive the third equation of motion (the one that relates velocity to position) using calculus [3]. We can't just reverse engineer it from a definition. We need to play a rather sophisticated trick.
The first equation of motion relates velocity to time. We essentially derived it from this derivative…
| dv | = a |
| dt |
The second equation of motion relates position to time. It came from this derivative…
| ds | = v |
| dt |
The third equation of motion relates velocity to position. By logical extension, it should come from a derivative that looks like this…
| dv | = ? |
| ds |
But what does this equal? Well nothing by definition, but like all quantities it does equal itself. It also equals itself multiplied by 1. We'll use a special version of 1 (dtdt) and a special version of algebra (algebra with infinitesimals). Look what happens when we do this. We get one derivative equal to acceleration (dvdt) and another derivative equal to the inverse of velocity (dtds).
| dv | = | dv | 1 | |
| ds | ds | |||
| dv | = | dv | dt | |
| ds | ds | dt | ||
| dv | = | dv | dt | |
| ds | dt | ds | ||
| dv | = | a | 1 | |
| ds | v |
Next step, separation of variables. Get things that are similar together and integrate them. Here's what we get when acceleration is constant…
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= |
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| v dv | = | a ds | ||||||||||||||
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= |
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| ½(v2 − v02) | = | a(s − s0) | ||||||||||||||
| v2 | = | v02 + 2a(s − s0) [3] | ||||||||||||||
Certainly a clever solution, and it wasn't all that more difficult than the first two derivations. However, it really only worked because acceleration was constant — constant in time and constant in space. If acceleration varied in any way, this method might become uncomfortably difficult.
motion with constant jerk
The method shown above works even when acceleration isn't constant. Let's apply it to a situation with an unusual name — constant jerk. No lie, that's what it's called. Jerk is the rate of change of acceleration with time.
| j = | da |
| dt |
This makes jerk the first derivative of acceleration, the second derivative of velocity, and the third derivative of position.
| j = | da | = | d2v | = | d3s |
| dt | dt2 | dt3 |
The SI unit of jerk is the meter per second cubed.
| ⎡ ⎢ ⎣ |
m/s3 | = | m/s2 | ⎤ ⎥ ⎦ |
| s |
An alternate unit is the g per second.
| ⎡ ⎢ ⎣ |
g | = | 9.80665 m/s2 | = 9.80665 m/s3 | ⎤ ⎥ ⎦ |
| s | s |
Jerk is not just some wise ass physicists response to the question, "Oh yeah, so what do you call the third derivative of position?" Jerk is a meaningful quantity.
The human body comes equipped with sensors to sense acceleration and jerk. Located deep inside the ear, integrated into our skulls, lies a series of chambers called the labyrinth. Part of this labyrinth is dedicated to our sense of hearing (the cochlea) and part to our sense of balance (the vestibular system). The vestibular system comes equipped with sensors that detect angular acceleration (the semicircular canals) and sensors that detect linear acceleration (the otoliths). We have two otoliths in each ear — one for detecting acceleration in the horizontal plane (the utricle) and one for detecting acceleration in the vertical place (the saccule). Otoliths are our own built in accelerometers.
The word otolith comes from the Greek οτο (oto) for ear and λιθος (lithos) for stone. Each of our four otoliths consists of a hard bone-like plate attached to a mat of sensory fibers. When the head accelerates, the plate shifts to one side, bending the sensory fibers. This sends a signal to the brain saying "we're accelerating." Since gravity also tugs on the plates, the signal may also mean "this way is down." The brain is quite good at figuring out the difference between the two interpretations. So good, that we tend to ignore it. Sight, sound, smell, taste, touch — where's balance in this list? We ignore it until something changes in an unusual, unexpected, or extreme way.
I've never been in orbit or lived on another planet. Gravity always pulls me down in the same way. Standing, walking, sitting, lying — it's all quite sedate. Now let's hop in a roller coaster or engage in a similarly thrilling activity like downhill skiing, Formula One racing, or cycling in Manhattan traffic. Acceleration is directed first one way, then another. You may even experience brief periods of weightlessness or inversion. These kinds of sensations generate intense mental activity, which is why we like doing them. They also sharpen us up and keep us focused during possibly life ending moments, which is why we evolved this sense in the first place. Your ability to sense jerk is vital to your health and well being. Jerk is both exciting and necessary.
Constant jerk is easy to deal with mathematically. As a learning exercise, let's derive the equations of motion for constant jerk. You are welcome to try more complicated jerk problems if you wish.
Jerk is the derivative of acceleration. Undo that process. Integrate jerk to get acceleration as a function of time. I propose we call this the zeroth equation of motion for constant jerk. The reason why will be apparent after we finish the next derivation.
| j = | da | |
| dt | ||
| da = | j dt | |
| a | t | ||
| ⌠ ⌡ |
da = | ⌠ ⌡ |
j dt |
| a0 | 0 |
| a − a0 = | jt | |
| [0] |
Acceleration is directly proportional to time when jerk is constant.
a ∝ t
Acceleration is the derivative of velocity. Integrate acceleration to get velocity as a function of time. We've done this process before. We called the result the velocity-time relationship or the first equation of motion when acceleration is constant. We should give it a similar name. This is the first equation of motion for constant jerk.
| a = |
|
|||
| dv = | a dt | |||
| dv = | (a0 + jt) dt | |||
| v | t | ||
| ⌠ ⌡ |
dv = | ⌠ ⌡ |
(a0 + jt) dt |
| v0 | 0 |
| v − v0 = | a0t + ½jt2 | ||
| [1] | |||
Velocity is proportional to time squared when jerk is constant.
v ∝ t2
Velocity is the derivative of displacement. Integrate velocity to get displacement as a function of time. We've done this before too. The resulting displacement-time relationship will be our second equation of motion for constant jerk.
| v = |
|
|||
| ds = | v dt | |||
| ds = | (v0 + a0t + ½jt2) dt | |||
| s | t | ||
| ⌠ ⌡ |
ds = | ⌠ ⌡ |
(v0 + a0t + ½jt2) dt |
| s0 | 0 |
| s − s0 = | v0t + ½a0t2 + ⅙jt3 | ||
| [2] | |||
Displacement is proportional to time cubed when jerk is constant.
s ∝ t3
Please notice something about these equations. When jerk is zero, they all revert back to the equations of motion for constant acceleration. Zero jerk means constant acceleration, so all is right with the equations we've created.
So where do we go next? Should we connect velocity-displacement (a third equation), acceleration-velocity (a fourth equation), acceleration-displacement (a fifth equation)? Look at that scary cubic equation for displacement. That can't be our friend.
The zeroth and third equations can be combined algebraically to make a fourth equation of motion for constant jerk. I won't show you any of the work — just the results, which should remind you of something. (It has the same form as the third equation of motion for constant acceleration.)
| a = | a0 + jt | [0] |
| + | ||
| v = | v0 + a0t + ½jt2 | [1] |
| = | ||
| [4] |
Acceleration is proportional to the square root of velocity when jerk is constant.
a ∝ √v
Algebra probably can't be used to generate the other two equations. Ask a mathematician if you need proof.
| v = | v0 + a0t + ½jt2 | [1] |
| + | ||
| s = | s0 + v0t + ½a0t2 + ⅙jt3 | [2] |
| = | ||
| v = | f(s) | [3] |
| a = | a0 + jt | [0] |
| + | ||
| s = | s0 + v0t + ½a0t2 + ⅙jt3 | [2] |
| = | ||
| a = | f(s) | [5] |
We can make it solvable if we just examine the special cases when s0 = v0 = a0 = 0. Then…
| v = | ½jt2 | [1′] |
| + | ||
| s = | ⅙jt3 | [2′] |
| = | ||
| v3 = | 368js2 | [3′] |
| a = | jt | [0′] |
| + | ||
| s = | ⅙jt3 | [2′] |
| = | ||
| a3 = | 6j3s | [5′] |
Which means that…
| v ∝ s2/3 | and | a ∝ s1/3 |
…when jerk is constant.
We also know that when jerk is zero, we should at least get back what we started with (our initial values). What we don't know is what all the other stuff should be in between. We should get equations that look something like these…
| v3 = | v03 + stuff + 368j(s2 − s02) | [3] |
| a3 = | a03 + stuff + 6j3(s − s0) | [5] |
This problem isn't done now, but I'm done with it for now.
| made up name | equation | relationship |
|---|---|---|
| 0th equation | a = a0 + jt | acceleration-time |
| 1st equation | v = v0 + a0t + ½jt2 | velocity-time |
| 2nd equation | s = s0 + v0t + ½a0t2 + ⅙jt3 | displacement-time |
| 3rd equation | v3 = v03 + stuff + 368j(s2 − s02) | velocity-displacement |
| 4th equation | a2 = a02 |
acceleration-velocity |
| 5th equation | a3 = a03 + stuff + 6j3(s − s0) | acceleration-displacement |
where…
| s = | position |
| s0 = | initial position |
| v = | velocity |
| v0 = | initial velocity |
| a = | acceleration |
| a0 = | initial acceleration |
| j = | jerk |
| t = | time |
motion with constant nothing
This page in this book isn't about motion with constant acceleration, or constant jerk, or constant snap, crackle or pop. It's about the general method for determining the quantities of motion (position, velocity, and acceleration) with respect to time and each other for any kind of motion. The procedure for doing so is either differentiation (finding the derivative)…
- The derivative of position with time is velocity (v = dsdt).
- The derivative of velocity with time is acceleration (a = dvdt).
or integration (finding the integral)…
- The integral of acceleration over time is change in velocity (∆v = ∫a dt).
- The integral of velocity over time is change in position (∆s = ∫v dt).
Here's the way it works. Some characteristic of the motion of an object is described by a function. Can you find the derivative of that function? That gives you another characteristic of the motion. Can you find its integral? That gives you a different characteristic. Repeat either operation as many times as necessary. Then apply the techniques and concepts you learned in calculus and related branches of mathematics to extract more meaning — range, domain, limit, asymptote, minimum, maximum, extremum, concavity, inflection, analytical, numerical, exact, approximate, and so on. I've added some important notes on this to the summary for this topic.
