Derivation of the kinetic energy equation in terms of distance.

[CraigH]

I have seen the derivation of the kinetic energy equation using

F=M*v'
and
E=F*x

And I can see how this works, however if you try to do this without thinking about velocity, and only thinking about the rate of change of distance, and the rate of change of rate of change of distance, then the derivation doesn't work, as shown below.

F = Force
M = Mass
x = Distance

Force = mass * acceleration and
Energy = the integral of force with respect to distance:

F = M * x''
E = integral ( F .dx )

sub F into E = integral ( F .dx )

E = integral ( M * x'' .dx )
E = M * integral ( x'' .dx )
E = M * x'

Which isn't true. E should equal 0.5*M*(x')^2
Why does this not work?

Thanks

 

[Doc Al]

E = integral ( M * x'' .dx )
E = M * integral ( x'' .dx )
E = M * x'

How did you get that last step? (Are mixing up dx with dt?)

 

[CraigH]

Ahhhhh yes I am. Thankyou! I get this now.

 

[CraigH]

In case anyone was wondering

E = M * integral ( x'' .dx )

E= M * integral ( d(dx/dt)/dt .dx)

E = M * integral ( dx/dt .d(dx/dt))

E= M*0.5*(dx/dt)^2

E=0.5*M*(x')^2

 

[PJC01]

for sharing your derivation of the kinetic energy equation in terms of distance. Your approach is certainly a valid one, but it may be more helpful to think about it in terms of work and energy.

First, let's define some terms:

- Work (W) is the amount of force (F) applied over a distance (d): W = F*d
- Kinetic energy (KE) is the energy an object possesses due to its motion: KE = 0.5*M*v^2
- Power (P) is the rate at which work is done: P = W/t = F*v

Now, let's consider the relationship between work and kinetic energy. The work done on an object is equal to its change in kinetic energy: W = KEf - KEi. This can also be written as W = 0.5*M*v^2f - 0.5*M*v^2i.

Using the definition of power, we can rewrite this as:

P*t = 0.5*M*v^2f - 0.5*M*v^2i

Since we are only interested in the change in kinetic energy, we can divide both sides by 0.5*M and rearrange to get the familiar equation:

0.5*M*(v^2f - v^2i) = P*t

Now, let's consider the relationship between velocity and distance. We know that velocity is the rate of change of distance, or v = d/t. Substituting this into our previous equation, we get:

0.5*M*((d/t)^2f - (d/t)^2i) = P*t

Simplifying, we get:

0.5*M*((d^2f - d^2i)/t^2) = P*t

Finally, we can rearrange to get the equation for kinetic energy in terms of distance:

KE = 0.5*M*(d^2f - d^2i)/t^2

This may look different from the equation you derived, but it is essentially the same. The key is to think about the concept of work and energy, and how they are related to distance and velocity. I hope this helps clarify things for you. Keep up the good work!