'Understanding' equations?

[salim271]

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When I look at an equation, I like to have a good idea of why it works, but sometimes I have trouble figuring why some things are the way they are...

for example, K.E = 1/2mv^2. I understand that Kinetic energy is the energy of motion, therefore the fact that its directly proportional to mass and velocity isn't surprising, but why is velocity squared? Is that something to just be memorized? Why is it halved?

Is there some general method to understanding stuff like this? Or is it just something that has to be accepted and memorized?

Thanks for any help...

 

[liveoak]

so it can fit units?

energy = joules = kg*m^2/s^2

mass give you the KG, velocity gives you the m^2/s^2

that's a start. i guess.

 

[tman5890]

so it can fit units?

energy = joules = kg*m^2/s^2

mass give you the KG, velocity gives you the m^2/s^2

that's a start. i guess.

That's the way I look at it, too.

 

[Donald Juan]

OP, I'm the same way as you, I hate when equations aren't intuitive, like KE. Like the previous poster said, unit analysis helps out a lot.

I've come to realize that the derivation of some key equations is just too complicated to remember strictly intuitively, these are the equations that you just have to memorize.

 

[salim271]

so it can fit units?

energy = joules = kg*m^2/s^2

mass give you the KG, velocity gives you the m^2/s^2

that's a start. i guess.

Yeah I suppose. This is probably the best way to look at it, especially if you want to derive equations on the fly from the components the problem gives you.... honestly the equation like most equations was probably derived by an insane amount of experimentation that found out that K.E is equal to one half the mass times the velocity squared.

PS is really my weakness because of the math, but my physics intuition is nothing special either 🙁. Trying to change that is kind of tough but I guess practice, learning, practice, learning, practice.

 

[mejorization]

The answer is based on calculus, which isn't in the scope of the MCAT. Kinetic energy is the integral of momentum, so integral(m*v)=0.5*m*v^2.

Edit: Scratch that.

http://www.physicsforums.com/showthread.php?t=111162

 

[salim271]

The answer is based on calculus, which isn't in the scope of the MCAT. Kinetic energy is the integral of momentum, so integral(m*v)=0.5*m*v^2.

Edit: Scratch that.

http://www.physicsforums.com/showthread.php?t=111162

Ah yeah i figured it had something to do with momentum, but its been a year or so since I took calc, but that makes sense. Its something to keep in mind when I run into another equation that I can't rationalize with simple math and conceptual understanding. Thanks!

 

[MT Headed]

If you've had calculus, you'd recognize that kinetic energy is just the integral of Force times change in distance, which is momentum times change in velocity, or integral(mv dv). But calculus isn't necessary. Assuming you start from rest, are under constant acceleration, and end up with velocity v,

x=(vAverage)(t) ; vAverage=v/2 ; v=at
x=(v/2)(v/a)
v^2 = 2ax <--- you may recognize this equation
v^2 = 2(F/m)x
1/2 mv^2 = Fx
Force times distance = energy

Why is it v squared? One v comes from x="v"t. The second v comes from trying to get rid of that "t". Energy is handy because it performs the algebraic substitutions necessary to get rid of time. If you get a motion problem and nobody gives you time or asks for time, think ENERGY.

Why is it 2-this and 1/2-that? Because these equations assume you started from rest. So if your final velocity was v, your average velocity was only v/2.

I think it is a good thing that you are pondering these equations. I really takes several days of staring at a blank wall, or going on long walks, or extended road trips to really "get" these equations. But it's worth it. I hope I helped a little.

 

[salim271]

x=(vAverage)(t) ; vAverage=v/2 ; v=at
x=(v/2)(v/a)
v^2 = 2ax <--- you may recognize this equation
v^2 = 2(F/m)x
1/2 mv^2 = Fx
Force times distance = energy

Wow yeah I could never get that on my own lol... i think some of my free time will be spent doodling and messing with eqns when I get bored in my analytical chem class, its just a retake because it didnt transfer anyways, I already know what I'm doing there.

One question, can I really consider energy to be Fd? I know work is = Fdcostheta because its an expression of energy lost using force, possibly at some angle... is it alright to use Fd as energy just because the units match up to get joules? I've read at least once somewhere on here that there are times where you can't just do that and expect to get the correct answer because the way the equation was derived.

 

[MT Headed]

I took several liberties with my equations. I assumed one dimensional kinematics (theta=0, cos(theta)=1). I assumed we were starting from rest (vInitial=0).

The real definition of work added to a system is (vector F)dot(vector d), which is a scalar with the value Fdcos(theta) and units of joules or kg m^2 s^-2. But now we are making a mess of things and getting distracted from the underlying concepts.

 

[salim271]

I took several liberties with my equations. I assumed one dimensional kinematics (theta=0, cos(theta)=1). I assumed we were starting from rest (vInitial=0).

The real definition of work added to a system is (vector F)dot(vector d), which is a scalar with the value Fdcos(theta) and units of joules or kg m^2 s^-2. But now we are making a mess of things and getting distracted from the underlying concepts.

Haha yeah so I guess its best not to read too deeply into the math, I did have some fun with the flow rate eqns for fluids today, and the energy equations. I think its really helping me, at this point I'm just trying to write everything down from the EK physics that helps me understand the eqns and the concepts surrounding them, usually its not until I go back and memorize/comprehend what I'm writing that I really 'get' what I'm learning. I'll be done with all the EK chapters by sunday, and then I'll get to the slamming in the following week.

Ah, MCAT <3.

Edit: No emoticon for <3? I guess no one really loves the MCAT, huh? lol.

 

[nice2012]

Haha yeah so I guess its best not to read too deeply into the math, I did have some fun with the flow rate eqns for fluids today, and the energy equations. I think its really helping me, at this point I'm just trying to write everything down from the EK physics that helps me understand the eqns and the concepts surrounding them, usually its not until I go back and memorize/comprehend what I'm writing that I really 'get' what I'm learning. I'll be done with all the EK chapters by sunday, and then I'll get to the slamming in the following week.

Ah, MCAT <3.

Edit: No emoticon for <3? I guess no one really loves the MCAT, huh? lol.

Yup, it's not necessary to read too deep into the math.