Derivative of energy is force?

In summary, the derivative of energy with respect to distance yields force, as shown by the example of kinetic energy, where the derivative of KE=1/2mv^2 with respect to distance, assuming constant mass, results in KE'= ma. This holds true for other forms of energy, such as potential energy stored in a spring. Additionally, Einstein's famous equation, E=mc^2, relates mass with energy, but does not mean that force is converted into mass as speed approaches the speed of light. Instead, the momentum, m0v/√(1 - v2/c2), increases indefinitely as speed approaches c. Therefore, doubling the force will only get twice as close to the speed of light. It is
  • #1
physicsisfeyn
2
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Why is the derivative of energy force? For example: KE=1/2mv^2 the derivative of which, assuming mass is held constant is KE'= ma. Why is this? What is the significance of this? What are some applications of this?

Additionally, Einstein's famous equation, E=mc^2, relates mass with energy. Does this mean as you approach the speed of light, that force is actually converted into mass? For example, if you have a mass of 1kg traveling with an acceleration of 299,000,000m/s and you double the force (which is 299,000,000N times 2) that some of the force is actually transformed into mass?
 
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  • #2
welcome to pf!

hi physicsisfeyn! welcome to pf! :smile:

(try using the X2 icon just above the Reply box :wink:)
physicsisfeyn said:
Why is the derivative of energy force? For example: KE=1/2mv^2 the derivative of which, assuming mass is held constant is KE'= ma.

nooo … derivative of momentum is force: d(mv)/dt = ma

d(1/2 mv2)/dt = mv dv/dt = mva

if you mean d(1/2 mv2)/dv, that's momentum, mv (and the meaning is that the laws of physics are invariant under a change of velocity of the observer, so differentiating the conservation of energy wrt v must still give a conservation equation, and the only one left is momentum!)

i don't see any way of getting ma from 1/2 mv2 :confused:
Additionally, Einstein's famous equation, E=mc^2, relates mass with energy. Does this mean as you approach the speed of light, that force is actually converted into mass?

no, e = mc² has nothing to do with the speed of light, the c² has no physical significance: it is simply the conversion factor between J and kg

if energy was measured in light-kilograms (in the same way that distance can be measured in light-years), then the conversion factor would be unnecessary, and the formula would be e = m :wink:
 
  • #3
Derivative of Energy or Work with respect to displacement s yields force. This is from the definition of work as integral of force over distance s and the basic theorem of calculus.

In the case of kinetic energy taking direvative wrt to s and have in mind that v=ds/dt we have dE/ds=mvdv/ds=mv(dv/dt)(dt/ds)=mavdt/ds=ma.
 
  • #4
F = dU/dx

and F = dP / dt
 
  • #5
Okay.

So am I wrong in thinking that mass increases with speed? If so, what happens when you approach the speed of light? If you're right at the barrier, say .9999c, and you double the force, what happens?
 
  • #6
you can keep increasing the momentum indefinitely

the momentum is m0v/√(1 - v2/c2), which "approaches infinity" as v approaches c …

loosely speaking, doubling the momentum gets you twice as close to c :wink:
 
  • #7
force times distance = energy
 
  • #8


tiny-tim said:
i don't see any way of getting ma from 1/2 mv2 :confused:

As somebody else said the derivative of energy with respect to distance is force

Keep it simple and assume the mass stays constant:

d/dx (m v^2 / 2)
= m/2 d/dx (dx/dt)^2
= m/2 dt/dx d/dt (dx/dt)^2
= m/2 dt/dx (2 dx/dt) d^2x/dt^2
= m d^2x/dt^2
= ma

For example the energy stored in a spring = 1/2 K x^2
and the force in the spring = d/dx (1/2 K x^2) = Kx
 

1. What is the relationship between energy and force?

The derivative of energy is force. This means that energy and force are directly related, and changes in energy can result in changes in force.

2. How is the derivative of energy related to motion?

The derivative of energy is related to motion through Newton's second law of motion, which states that force is equal to the rate of change of an object's momentum. This means that the derivative of energy, which is force, is directly related to the motion of an object.

3. Why is the derivative of energy important in physics?

The derivative of energy is important in physics because it helps us understand the relationship between energy and force, which is essential in understanding the motion and behavior of objects. It also allows us to study and predict the effects of changes in energy on force and motion.

4. Can the derivative of energy ever be negative?

Yes, the derivative of energy can be negative. This would mean that the force acting on an object is in the opposite direction of its motion, resulting in a decrease in its energy.

5. How is the derivative of energy calculated?

The derivative of energy is calculated using calculus, specifically the derivative formula. This involves taking the derivative of the energy function with respect to time, which results in the force function.

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