Is Kinetic Energy Derived from an Integral and Work Done?

In summary, the conversation discusses the relationship between Hookes law and the energy stored in an elastic rope. It is shown that the energy stored can be calculated using an integral, similar to the equation for kinetic energy. The question is raised whether the 1/2 factor in the equation for kinetic energy is also the result of an integral and if it is related to work done with respect to velocity. The conversation concludes with an explanation of dW as the work done.
  • #1
RoryP
75
0
Hi all,
I was doing an A-level Mechanics paper the other day and one of the quesitons was to show that, starting with Hookes law T=(lambda)(x)/(l), show that the energy stored in an elastic rope is (lambda)(e2)/(2l).
This was ok, I just said that energy stored would be equal to the sum of the work done stretching the rope a small distance δx and as δ tended to zero it would be dx. Which could be re-written as integral with limits e and 0 dx.
Which leads to the equation for elastic potential energy.
After doing this i realized that kinetic energy is in a similar form, i.e power of 2 and has a multiplying factor of 1/2 which leads me to my question, is 1/2mv2 the result of an integral? Has it also got something to do with work done? but with respect to v?
Any help would be greatly appreciated!
 
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  • #2
dW = f*dx = m*a*dx = m*dv/dt*dx = m*v*dv ( since dx/dt = v)
Take the integration between 0 to v.
 
  • #3
ok brilliant thanks! what is dW? respect to work done?
 
  • #4
Yes.
 

What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion.

What is the formula for kinetic energy?

The formula for kinetic energy is KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

How is kinetic energy derived?

Kinetic energy is derived by integrating the equation for work, W = F * d, where F is the force applied to the object and d is the distance it travels.

What is the relationship between kinetic energy and mass?

The kinetic energy of an object is directly proportional to its mass. This means that as mass increases, kinetic energy also increases.

How does kinetic energy relate to potential energy?

Kinetic energy and potential energy are two forms of mechanical energy. Kinetic energy is the energy of motion, while potential energy is the energy an object has due to its position or state. In some cases, potential energy can be converted into kinetic energy, and vice versa.

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