How is Work Related to Velocity?

[Eeeff]

Homework Statement

Help. If
w = ∫F dx
where w is work, f is force and x is displacement, and also
F = k₁v²
where force is a quadratic function of velocity v times a constant k₁,
what would be work with regards to velocity?

Homework Equations

More info if needed.
v = ∫a dt
a = k₂ - k₁v²/m
k₂ is another constant representing the initial acceleration, and m is the mass of the object.

The Attempt at a Solution

I have some attempts but I am just clueless.

Thanks.

 

[Arman777]

I have an idea.F is a funtion of v so W=∫F(v)dx the main purpose should be solve this integral but for that it must be in the same "thing".Like ∫tdt or ∫vdv etc.
Using a = k2 - k1v2/m and some equations you can find it.

 

[kuruman]

a = k2 - k1v2/m

I don't see where you got this. If F = ma and F = k₁v², what is a?

 

[haruspex]

There is a very useful formula for when you don't care about the time of the motion:
$a=\frac{dv}{dt}=\frac{dx}{dt}\frac{dv}{dx}=v\frac{dv}{dx}$.

 

[kuruman]

There is also the work-kinetic energy theorem.

 

[Eeeff]

I don't see where you got this. If F = ma and F = k₁v², what is a?

Let me clarify, F_ext = k₁v². F_net = ma = mk₂ - F_ext.

 

[Eeeff]

Let me clarify, F_ext = k₁v²
F_net = ma = mk₂- F_ext

 

[kuruman]

Let me clarify, Fext = k₁v². Fnet = ma = mk₂ - Fnet.

How many forces are there? If there is only one, the net force, i.e. the sum of all the forces, is that one force.

 

[Eeeff]

How many forces are there? If there is only one, the net force, i.e. the sum of all the forces, is that one force.

Sorry for not making this clear. There is force_ext1 = k₁v² that is acting against the object and constant force_ext2 = mk₂ acting on the object.

 

[kuruman]

Sorry for not making this clear. There is forceext1 = k₁v² that is acting against the object and constant forceext2 = mk2 acting on the object.

OK. The work done by which force is the question asking you to calculate?

I know you are new to PF, but for future reference, please post the problem exactly as it is given to you. It will avoid misunderstandings and your question will be answered more quickly.

 

[Eeeff]

OK. The work done by which force is the question asking you to calculate?

I know you are new to PF, but for future reference, please post the problem exactly as it is given to you. It will avoid misunderstandings and your question will be answered more quickly.

I wish to know the work done by F ext1. Anyway, why would I want to know the force done by the constant force, when it is just W = Fd.

 

[kuruman]

Anyway, why would I want to know the force done by the constant force, when it is just W = Fd.

It's not what you want to know, it's what the problem is asking you to answer. Actually, for this problem, finding the work done by the constant force is part of the answer. You can use the hint provided by haruspex (post#4) or by me (post #5); they are basically the same idea for tackling the problem.