Justification for differential manipulation in work-KE theorem proof

In summary: The substitution "dx=vdt" simply allows us to change the limits of integration from x to t, making the integral easier to solve. This substitution is justified by the chain rule/substitution in calculus. In summary, the mathematical justification for the substitution "dx=vdt" in the work-KE theorem is based on the chain rule/substitution in calculus, which allows us to change the limits of integration from x to t. This substitution is valid because v is a function of both time and position, and can be treated as such in the integral.
  • #1
shooba
9
0
Hi, this question is about the mathematical justification for a physics topic so hopefully this is the right forum.
All the proofs of the work-KE theorem I have found go something like this:
W= int(F)dx from x1 to x2
= m(int(dv/dt))dx from x1 to x2
= m(int((dv/dt)v)dt from t1 to t2
= (m/2)(vf^2-vi^2) by the chain rule/substitution
My question is regarding the substitution "dx=vdt," and the associated change of variables from x to t, which is obviously true but how do you justify it from the substitution rule or something else not involving differentials? (I'm sure that expression makes perfect sense when it comes to differential forms but I have not learned those yet)
 
Physics news on Phys.org
  • #2
hi shooba! :smile:

(have an integral: ∫ and try using the X2 and X2 icons just above the Reply box :wink:)
shooba said:
All the proofs of the work-KE theorem I have found go something like this:
W= int(F)dx from x1 to x2
= m(int(dv/dt))dx from x1 to x2
= m(int((dv/dt)v)dt from t1 to t2
= (m/2)(vf^2-vi^2) by the chain rule/substitution

no, that's wrong, there's no need for a change in variable from dx to dt

it should only use the https://www.physicsforums.com/library.php?do=view_item&itemid=353" …

m(∫(dv/dt))dx from x1 to x2

= m(∫(dv/dx)(dx/dt))dx from x1 to x2 (from the chain rule)

= m(∫(dv/dx)v)dx from x1 to x2

= m(∫(dv/dx)v)dx from x1 to x2

= m(∫(d(v2/2)/dx))dx from x1 to x2

= (m/2)(v22-v12)
 
Last edited by a moderator:
  • #3
Wow thanks, that's a lot clearer than the other derivations!
Just one question; when we write dv/dx we are treating velocity as a function of x; does this somehow conflict with the fact that velocity is the time derivative of x and thus a function of t? Is "v" somehow a function of both? (this might be more of a physics question)
 
  • #4
hi shooba! :smile:

(just got up :zzz: …)
shooba said:
… Is "v" somehow a function of both? (this might be more of a physics question)

no, it's a maths question …

suppose you're in a car looking at the speedometer, and you want to draw a graph of the speed,v …

you can look at your watch, and draw a graph of v against t, or you can look at the milometer, (or mileposts), and draw a graph of v against x …

the speedometer doesn't know how you're graphing it …

all it produces is v :wink:​

v is a function of whatever we want it to be
 

What is the work-KE theorem and why is it important in scientific research?

The work-energy theorem is a fundamental principle in physics that states the total work done on an object is equal to the change in its kinetic energy. It is important in scientific research because it allows scientists to understand and calculate the relationship between work and energy in various systems.

What is differential manipulation and how does it relate to the work-KE theorem proof?

Differential manipulation refers to the process of manipulating equations or variables in a mathematical proof to arrive at a desired result or conclusion. In the context of the work-KE theorem proof, differential manipulation is often used to simplify equations and make them more manageable.

Why is justification needed for differential manipulation in the work-KE theorem proof?

Justification is needed for differential manipulation in the work-KE theorem proof because it ensures that the manipulations performed are mathematically valid and do not alter the fundamental principles of the theorem. Without proper justification, the proof may be considered invalid and unreliable.

What are some common justifications used in differential manipulation for the work-KE theorem proof?

Some common justifications used in differential manipulation for the work-KE theorem proof include the use of algebraic rules and properties, such as the distributive property and the commutative property. Physical principles and laws, such as the conservation of energy and the principle of virtual work, may also be used to justify certain manipulations.

How does differential manipulation in the work-KE theorem proof contribute to the overall understanding of the theorem?

Differential manipulation in the work-KE theorem proof allows scientists to break down complex equations and analyze them step by step, leading to a deeper understanding of the theorem and its applications. By manipulating equations, scientists can also gain insights into the relationships between different variables and how they affect the overall outcome of the theorem.

Similar threads

I Second derivative, chain rules and order of operations
    • Calculus
Replies
6
Views
2K
I Gaussian integral by differentiating under the integral sign
    • Calculus
Replies
19
Views
3K
I On convolution theorem of Laplace transform: Schiff
    • Calculus
Replies
4
Views
720
Differentiate the given integral
    • Calculus and Beyond Homework Help
Replies
15
Views
774
B Product rule OR Partial differentiation
    • Calculus
Replies
4
Views
3K
I On differentiability and Fourier coefficients (Vretblad's text)
    • Topology and Analysis
Replies
4
Views
353
Integration operation, and its relation to differentials
    • Calculus
Replies
11
Views
1K
MHB Sample problems for MVT/Integral proofs (elementary level calc)
    • Calculus
Replies
0
Views
846
I The far reaching ramifications of the work-energy theorem
    • Classical Physics
Replies
31
Views
2K
A Time differentiation of fluid line integrals
    • Calculus
2
Replies
45
Views
4K

Hot Threads

Recent Insights

Back
Top