The equivilence of Power and Kinetic Energy

In summary, the conversation is discussing the relationship between the power delivered by a force and the rate at which a particle is changing. It is shown that this can be represented by the equation P = m\vec{a} \bullet \vec{v} and later simplified to 2\vec{a} \bullet \vec{v} = \frac{d (|{v}|^2)}{dt}. The concept of the chain rule is used to explain the presence of the 2 in the equation. The speaker is seeking clarification on this concept.
  • #1
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This isn't homework, just be going over a few concepts.

I'm trying to show that the power delivered by a force equals the rate at which the particle is changing.

Now P = [tex]\vec{F}[/tex] [tex]\bullet[/tex] [tex]\vec{v}[/tex]
= m[tex]\vec{a}[/tex] [tex]\bullet[/tex] [tex]\vec{v}[/tex]
= m[tex]\vec{v}[/tex] [tex]\stackrel{\delta}{\delta t}[/tex][tex]V^2[/tex]

This book is now telling me that the above line = 2[tex]\vec{a}[/tex] [tex]\bullet[/tex] [tex]\vec{v}[/tex]whyyyyy?
 
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  • #2
I can't really read what you wrote...

[tex]P = \vec{F} \cdot \vec{v}[/tex]

[tex]P = m \vec{a} \cdot \vec{v}[/tex]

[tex]P = m \frac{d \vec{v}}{dt} \cdot \vec{v}[/tex]

[tex]P = m \frac{d}{dt}\frac{||{v}||^2}{2}[/tex]

But after that you get that

[tex]2 \vec{a} \cdot \vec{v} = \frac{d ||{v}||^2}{dt}[/tex]
 
  • #3
Feldoh said:
I can't really read what you wrote...

[tex]P = \vec{F} \cdot \vec{v}[/tex]

[tex]P = m \vec{a} \cdot \vec{v}[/tex]

[tex]P = m \frac{d \vec{v}}{dt} \cdot \vec{v}[/tex]

[tex]P = m \frac{d}{dt}\frac{|{v}|^2}{2}[/tex]

But after that you get that

[tex]2 \vec{a} \cdot \vec{v} = \frac{d (|{v}|^2)}{dt}[/tex]

In your 4th line, where does that 2 come from? I still don't understand the jump from v to a, shouldn't it be only 2a? Not 2a x V?
 
  • #4
In the fourth line comes from the chain rule, which is also the reason you get 2a dot v.

[tex]\frac{d}{dt} \frac{v(t)^2}{2} = (2 v(t)) \frac{d}{dt}\frac{v(t)}{2}[/tex]
 
  • #5
Oh snap, totally forgot about that. Thank you.
 

Related to The equivilence of Power and Kinetic Energy

What is the definition of power?

Power is the rate at which work is done or energy is transferred. It is measured in watts (W) or joules per second (J/s).

What is the definition of kinetic energy?

Kinetic energy is the energy an object possesses due to its motion. It is dependent on the mass and velocity of the object and is measured in joules (J).

What is the difference between power and kinetic energy?

The main difference between power and kinetic energy is that power is a measure of how fast work is done or energy is transferred, while kinetic energy is a measure of the total energy an object has due to its motion.

How are power and kinetic energy related?

Power and kinetic energy are directly related, as power is equal to the change in kinetic energy over time. This can be represented by the equation P = ΔKE/Δt, where P is power, ΔKE is change in kinetic energy, and Δt is change in time.

How does the concept of the equivalence of power and kinetic energy apply to real-world situations?

In real-world situations, the equivalence of power and kinetic energy can be seen in various forms of motion, such as a moving car or a rotating wind turbine. In these cases, the power being generated is directly proportional to the kinetic energy of the object in motion.

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