Kinematics

11th grade Physics.

Memorize and understand these, and you'll pass the chapter test.

V_f = V_i + at
V_f² = V_i² + 2ax
X = Vit + (1/2)at²
g = 9.81

note, g = relative gravitational force, an object falling has an a of g.

Please note that I have only a superficial understanding of complex physics; this is simple physics (and in truth, only some equations associated with some simple physics) from someone with a simple understanding. I'd love to hear any advice or corrections from the experts!

The equations for position, velocity, and acceleration are some of the first and most basic taught to any physics student. For me - a dabbler without the time or talent for the complex stuff - they're the most beautiful bridge between thought and reality I've ever seen. Three equations: all related mathematically, expressions of each other; all related conceptually, expressions of the physical reality in which we exist. You can derive any one from any of the others, or from the motion you see around you; you can predict what you'll see by their workings. They're all one and the same - the concepts, the equations, the tiny facet of Being they represent - all wrapped up in a packet of understanding. It's Motion in form of Reason.

Plus, if you ever need to figure out how long it'll take an egg to hit the ground or when a car that's braking slower than the one in front of it is going to crash, nothing serves better!

Here are a few concepts you need to understand in orders to use these equations:
position: position is, simply, where an object is along the given axis. This is in units of distance, usually meters (m).
velocity: velocity is the rate at which an object's position is changing, i.e. how fast it's moving. Velocity is given in units of distance per unit of time, which is usually one second (s). Thus, the position of an object with a velocity of 5 m/s is changing by 5m every second. You should use the same unit of distance for your velocity values as you use for your position values, or you'll have to do a lot of extra conversion. acceleration: acceleration is the rate at which an object's velocity is changing - basically, how quickly it's speeding up or slowing down. Acceleration is given in units of velocity per unit of time; that is, units of distance per unit of time, per unit of time. So, an object that is speeding up by 5m/s every second has an acceleration of 5m/s². Again, you should use the same units as you do for position and velocity, or face painful complication.

This, also, is key: remember that all of these equations are scalar, which means we only deal with one axis (or dimension) at a time; however, velocity and acceleration are vectors. That means that they have a magnitude and a direction, both which you have to take into account. Imagine it this way: you're thinking about where a car is on a certain road, but it turns left before it starts to drive. It has an acceleration and a velocity, but with respect to your road both are zero. Objects can have positions measured along more than one axis as well, so if you're dealing with things occurring in more than one dimension, you need to use some trigonometry to break all of your vectors down into x, y, and z components, and solve for each component separately. SOH CAH TOA!

Here are the equations:

Position:
S = S₀ + v₀t + 1/2at²
S₀ is the initial position of the object in question (that is, the object's position when time = 0.)
v₀ is the initial velocity of said object.
t is time.
a is the object's acceleration.

Now, as you'll recall, velocity is the rate of change of position. If you're mathematically inclined (or just had your calculus drilled into you really, really thoroughly,) you know that means derivative. So, it's no surprise that one equation for velocity is the derivative of the one for position (and thus, that the position equation is the integral of the velocity equation!) Too cool.

Velocity:
v = v₀ + at
v₀ is the object's velocity at time 0.
a is the object's acceleration.
t is, once again, time.

(There's also another, which you can use if you don't have (or don't care) about time: v² = v₀² + 2aS. If only it were so easy to remove the tyranny of time from the rest of life!)

Let's repeat our logic from above: if you can derive velocity from position because velocity is the rate of change of position, you can do just the same with acceleration. It's the rate of change of velocity, so its equation is the derivative of the velocity equation:

Acceleration:
a = a
a is, of course, acceleration.

Now, that's kind of a let-down! But if you think about it, this tautology is the simple truth that the other equations rest upon, as you get them by integrating it. So, it's not so useless after all, at least conceptually.

What do you do with these equations? Gain a greater understanding of your world - and solve story problems, of course! So, imagine you're 100m off the ground, and you drop an egg - how long will it take to land? Let's find out.

First, you need to pick an equation. Since we're dealing with an issue that involves position, we'd better pick an equation that does too, which leaves only one possibility (S = S₀ + v₀t + 1/2at².)

Now, let's figure out what we know. In this case, we already know the final position of the egg - ground level. Let's call that 0. So:
s = 0.
We also know how high up we are, which is the egg's initial position. So:
s₀ = 100m.
We're just dropping the egg, so:
v₀ = 0.
It'd be easy to put in a value that'd represent throwing the egg, but that would just be unnecessarily destructive, wouldn't it?
Anyway, we also know the acceleration due to gravity, since that's a constant: 9.8 m/s. Now, we're saying that ground level is 0 and counting up as we go up, but the acceleration due to gravity is clearly in the opposite direction (take a look at the excellent Sign Convention article for some good advice on how to decide this sort of thing.) So:
a = -9.8m/s² (Note: if we want to be really pointy-headed, we could adjust this for air resistance!)

Now, plug everything in and do some solving:

1. 0m = 100m + 0m/sts + 1/2(-9.8m/s²)ts²
   
2. 0m = 100m + 0m + -4.9m/s²t²s²
   
3. 4.9m/s²t²s² = 100m
   
4. t²s² = 20.408s² (we divided both sides by 4.9m/s² here)
   
5. ts = 4.517s.

Notice how the units worked out just right? That's a sign that you did everything correctly. If you get t in units of meters, for example, you can be sure something is wrong. Putting the units in the equation complicates an easy math problem, though, and you can always check your answer by plugging the value you get back into the equation and solving it. So, let's do that instead.

1. 0 = 100 + 0t + 1/2(-9.8)t²
   
2. 0 = 100 + 0 + -4.9t²
   
3. 4.9t² = 100
   
4. t² = 20.408 (we divided both sides by 4.9 here)
   
5. t = 4.517.

Now, if we solve 0 = 100 + 0(4.517) + 1/2(-9.8)(4.517)², we get 0 = .023: we're a little off due to rounding errors, but correct. So now you know that if you drop an egg from 100m up, it'll take almost exactly four and a half seconds to hit. And do you know what? That's always what happens. Now you know why, and understand how it all works. That's what makes this stuff worth knowing, and what makes it beautiful.

Thanks to Kidas for suggesting that I add the second velocity equation, and to stupot, who called me on my inadequate treatment of vectors. Thanks also to Anark who points out that the mathematical relationships between the equations don't have any profound physical significance (like E=MC², which does say something profound about the universe we live in.) The derivative of the position equation describes velocity because velocity is defined as the rate of change of position, not for any deeper reason. For someone (like me) who likes the whole thing because of its pure "make-senseness," though, that's enough to make it seem cool!

I searched as thoroughly as I could and couldn't find these anywhere else. If they do exist elsewhere in a writeup I couldn't find, please let me know so I can either move this to the right place, or remove it if it's totally redundant. Thanks!

  Being in my second year of advanced placement physics, I have had all of last year in physics to establish a fair grasp on the equations for velocity, position, and acceleration. There are basically two types of equations (velocity & distance) that are used when solving for velocity, position and/or acceleration. These two types of equations have two forms (when the object is at rest & when it is already moving).

Distance:
Starting from rest
ΔD=.5AΔT²
With initial velocity
ΔD=V_iΔT + .5AΔT²

Velocity:
Starting from rest
V_f=AΔT
V_f²=2AΔD
With initial velocity
V_f=V_i + AΔT
V_f²=V_i² + 2AΔD
Acceleration:
A=D/T

Legend:
V_fstands for final velocity. (the velocity reached after so much time has passed)
A is the objects acceleration
D represents distance or displacement
T is the amount of time in question
Δ Delta stands for "change of". The Greek letter (delta) designates a change in the quantity of the variables D and T

  Now that you have all the equations and know when to use each one, go solve some problems and impress your friends by telling them how fast a penny is traveling when dropped from the Sears Tower (or some other very tall structure)

_{Disclaimer: These equations will only give you an idea of the distance, velocity or acceleration of a object in real life. The only time when these equations work perfectly is in Utopian Physics.}

Note: This writeup was re-noded @ the request of Darksnake... damn quitter.

GTKY? A true story nontheless

There are some equations you just can't forget.

Groupwork is something you would probably rather not deal with in college, but it's a necessary evil when incoming freshmen are forced to take one natural science course and the lab space doesn't allow people to work alone. (The one common thread to unite lower and higher education systems is that you have to deal with people that you dislike on a fairly regular basis.)

When I was one of said incoming freshmen, on the second week of class we were doing a lab assignment to approximate the gravitational constant, g, by rolling a baseball-sized smooth rubber ball down a ramp angled at 45 degrees.
By slightly manipulating the kinematics equations for a vector, one could approximate the constant.

"..or, we could just write down 9.8 m/sec^2 and be done with it," added unproductive lab partner number one. "I'm a philosophy major," he added helpfully, stretching out the word philosophy as long as possible, as if the additional length of the word gave his choice of major more credibility.

"Physics is pretty cool, though, man," my other lab partner said. His bony, pale fingers curled around the ball as he gripped it on the flameproof tabletop. His gaze was pensive and vaguely threatening, as if the fate of the world rested upon his ability to roll the lime green ball to its full extent.

I didn't care enough to intervene, so instead I opted to focus my energy on looking busy by reading the prelab.

"Seriously though - this is bullshit. Physics is bullshit."

"I wasn't aware all philosophy majors were so insightful." (with obligatory mocking emphasis on the word philosophy, of course)

"..kinematics is bullshit," he continued.

Through the corner of my eye, I saw the ball float through the air and interrupt his rant by impacting with his forehead and bouncing unceremoniously under a stool. There was an awkward pause as he was cut off mid-sentence, and the two freshmen looked at each other for a brief moment. The first looked on incredulously as the second knelt down to retrieve the ball, before standing back up and stating, "V equals V-O plus A-T.
..so can we roll the damn ball now?"

..and we did.

Kin`e*mat"ics (?), n. [Gr. (),() motion, fr. to move.] Physics

The science which treats of motions considered in themselves, or apart from their causes; the comparison and relation of motions.

Kinematics forms properly an introduction to mechanics, as involving the mathematical principles which are to be applied to its data of forces.

Nichol.

 

© Webster 1913.