Error in equation for kinetic energy

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Discussion Overview

The discussion revolves around the derivation of the kinetic energy equation, specifically addressing a claim that the work done by a constant force results in an expression of 2mv² instead of the standard (1/2)mv². Participants explore the implications of constant acceleration and the definitions of average and final velocity in this context.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that using F = ma leads to the conclusion that F*s = 2mv², questioning the validity of the (1/2) factor in the kinetic energy formula.
  • Another participant challenges the initial claim by stating that the object does not travel a finite distance during an infinitesimal time interval, suggesting a misunderstanding of the concepts involved.
  • A participant attempts to clarify that the average velocity should be used in the work-energy calculation, noting that the final velocity is twice the average velocity for constant acceleration from rest.
  • Further clarification is provided that for constant acceleration, the relationship a*dt = v should be applied, rather than a*dt = 2v.
  • One participant emphasizes that the work done should yield (1/2)mv² when the correct average velocity is used in the calculations.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of velocity in the context of work done by a constant force. There is no consensus on the derivation of the kinetic energy equation, as multiple competing explanations and corrections are presented.

Contextual Notes

Participants highlight potential misunderstandings regarding the definitions of average and final velocity, as well as the implications of constant acceleration on the calculations. The discussion remains focused on the mathematical relationships without resolving the underlying disagreements.

Hymne
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Okey, let's say we got an object with no start velocity (V_i = 0) which is being pushed by a constant force F, under a certain distance s (during a time intervall dt).
We get through F = ma that
F*s = ma*s = ma*v*dt
We also know that when a is constant (which it is due to the constant force) we got:
a*dt = 2v
which gives us that the work, F*s, equals
2mv^2 instead of (1/2)mv^2
What is wrong?
 
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The object does not travel a finite distance s during the infinitesimal time dt. You are mixing your concepts here.

F*ds = m*a*v*dt

Try taking it from here.
 
Ah, dt says that t_2 - t_1 approaches zero. But let's just use (t_2 - t_1), I still get it wrong :(
F*s = m*a*v*(t_2 - t_1) = m*v*2v = 2mv^2

I assume that over our given time and with our constant force we have a * (t_2 - t_1) = 2*v.
 
Did you know that

a=dv/dt?
 
Hymne said:
Okey, let's say we got an object with no start velocity (V_i = 0) which is being pushed by a constant force F, under a certain distance s (during a time intervall dt).
We get through F = ma that
F*s = ma*s = ma*v*dt
We also know that when a is constant (which it is due to the constant force) we got:
a*dt = 2v
which gives us that the work, F*s, equals
2mv^2 instead of (1/2)mv^2
What is wrong?

You're using "v" to mean the average velocity (= s/dt)

When we say that work = (1/2) m v^2, "v" is the final velocity.

The final velocity is twice the average velocity, they are not equal.
 
Hymne said:
Okey, let's say we got an object with no start velocity (V_i = 0) which is being pushed by a constant force F, under a certain distance s (during a time intervall dt).
We get through F = ma that
F*s = ma*s = ma*v*dt
You here assume that s = v*Δt, but that should be s = vave*Δt, where vave is the average speed. For constant acceleration starting from rest, vave = v/2, where v is the final speed. So really:
F*Δs = ma*Δs = ma*(v/2)*Δt

We also know that when a is constant (which it is due to the constant force) we got:
a*dt = 2v
No, for constant acceleration, starting from rest: a*Δt = v.
which gives us that the work, F*s, equals
2mv^2 instead of (1/2)mv^2
Nope, you get:
F*Δs = ma*Δs = ma*(v/2)*Δt = m*(v/2)*a*Δt = 1/2mv^2.

As expected. :wink:
 

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