Bored and curious about kinetic energy

[mburt]

So I was really bored today, and while pondering to myself I wondered why this isn't a valid formula for kinetic energy (say for a laterally moving truck):

E = md_x² / t²

So I did a little test example, say a moving truck with:

m = 2000 kg
d_x = 150 m
t = 10s

And plugged it in the formula and got 450 000 kg m²/s² (J).

After remember the ACTUAL kinetic energy formula (Ek = 0.5mv^2), I was wondering where on Earth the 0.5 came from? In reality the truck has a kinetic energy of 225 000 J

So my question is simply this: (probably a basic one) Why is there a 1/2 constant in the kinetic energy formula?

 

[AlephZero]

You need the 1/2 to be consistent with the law of conservation of energy.

Think about a falling object (ignoring air resistance).

The force acting on it is its weight = mg

The acceleration is constant = g

If it start from rest, you know that at time t

v = gt
x = 1/2 g t^2

The work done by its weight = force times distance = (mg) (1/2 g t^2)

The kinetic energy = 1/2 m (gt)^2

And those two quantities must be equal.

 

[timthereaper]

You could also think of it as a consequence of the integrals in determining the work needed to accelerate the particle.

http://en.wikipedia.org/wiki/Kinetic_energy#Derivation

 

[mburt]

If it start from rest, you know that at time t

v = gt
x = 1/2 g t^2

Is this considered integration? It looks like you've worked backwards to find the original equation for the derivative v = gt. I've only done introductory calculus btw, and still learning.

And what would x represent here? EDIT: Never mind! I guess it means a distance, considering the units work out to meters! This makes sense too, considered F = mg, and Ek must be in J = Nm... so the distance aspect is (1/2 gt^2) and the "N" aspect = mg, so you must multiply them... awesome!

Thanks