Is this a constant or variable force?

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Discussion Overview

The discussion revolves around whether the force acting on a vehicle stopping due to kinetic energy and drag is constant or variable. Participants explore the implications of this distinction on the formulation of differential equations related to the vehicle's stopping distance and energy loss. The conversation includes mathematical reasoning and attempts to derive equations based on the vehicle's dynamics.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that the net force on the vehicle is a function of velocity and time, suggesting it is non-constant.
  • Others present a differential equation relating the change in kinetic energy to the net force and displacement, questioning whether this formulation can apply to a variable force.
  • A participant expresses uncertainty about using a formula for constant force when the force is variable, indicating a potential misunderstanding of the underlying physics.
  • Another participant suggests a more straightforward approach by defining force in terms of acceleration and solving the resulting ordinary differential equation (ODE).
  • Some participants discuss the numerical solution using an Euler method for the differential equation, raising questions about the consistency of results between variable and constant force scenarios.
  • There is a mention of integrating a derived equation to find the maximum stopping distance, with a focus on the constants involved in the solution.

Areas of Agreement / Disagreement

Participants express differing views on whether the force is constant or variable, with no consensus reached. The discussion remains unresolved regarding the implications of this distinction on the equations used to describe the vehicle's stopping behavior.

Contextual Notes

Participants highlight limitations in their approaches, such as the dependence on definitions of force and the assumptions made when applying formulas for constant force to a variable force scenario. There are unresolved mathematical steps in deriving the equations.

revolution200
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If a vehicle is stopping its change in Kinetic energy is equal to the force times displacement or distance as it is traveling in a straight line and parallel to displacement.

If the force is F(net) = kinetic + drag

F(net) = umg + 1/2*A*rho*v^2

is this force constant or variable

If it is constant then the Kinetic energy (T) is

delta(T) = F(net) delta(s)

If it is variable then it is the formula for variable force

What I'm hoping to get is a differential equation of the form

dT/ds=F(net)
 
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revolution200 said:
If a vehicle is stopping its change in Kinetic energy is equal to the force times displacement or distance as it is traveling in a straight line and parallel to displacement.

If the force is F(net) = kinetic + drag

F(net) = umg + 1/2*A*rho*v^2

is this force constant or variable
The force is clearly a function of velocity and hence time and is therefore, non-constant.

What exactly are you trying to solve for?
 
I have solved it for distance.

dT/ds = B+CT where v^2=2T/m

solving I got

s = ln(B+CT) - ln(B) = ln(B+CT/B)

looking back through definitions I have noticed that I have used a formula for constant force when it isn't but I'm not sure
 
revolution200 said:
I have solved it for distance.

dT/ds = B+CT where v^2=2T/m

solving I got

s = ln(B+CT) - ln(B) = ln(B+CT/B)

looking back through definitions I have noticed that I have used a formula for constant force when it isn't but I'm not sure
I'm not sure why/how you're invoking energy here. And no, as I said above you cannot treat the force as constant.

A must more straightforward method would be to simply note the definition of force in terms of acceleration and solve the resulting ODE.
 
I have a vehicle stopping. I have already solved this issue and am writing the result up. I have descibed the stopping distance of the vehicle as a differential equation that describes the rate that the vehicle is losing energy. When the vehicle has stopped its Kinetic energy is zero.

The solution I have for this is

Where Work = change in Kinetic energy = Force * displacement

dT = F(net)ds

dT/ds = F(net)

where F(net) = umg + drag equation http://en.wikipedia.org/wiki/Air_resistance

for the v^2 term in the drag equation I substitute V^2 = 2T/m

so

F(net) = B + CT

the frictional force plus the drag force in terms of kinetic energy

I have also calculated a numerical solution using an Euler method that iterates down using the differential equation.

The problem I have had is do I get the same result for variable force as I do with constant force
i.e.
dT = F(net)ds
 
revolution200 said:
I have a vehicle stopping. I have already solved this issue and am writing the result up. I have descibed the stopping distance of the vehicle as a differential equation that describes the rate that the vehicle is losing energy. When the vehicle has stopped its Kinetic energy is zero.

The solution I have for this is

Where Work = change in Kinetic energy = Force * displacement

dT = F(net)ds

dT/ds = F(net)

where F(net) = umg + drag equation http://en.wikipedia.org/wiki/Air_resistance

for the v^2 term in the drag equation I substitute V^2 = 2T/m

so

F(net) = B + CT

the frictional force plus the drag force in terms of kinetic energy

I have also calculated a numerical solution using an Euler method that iterates down using the differential equation.

The problem I have had is do I get the same result for variable force as I do with constant force
i.e.
dT = F(net)ds

You write it as an ODE.

[tex]dT/ds = -b - cT[/tex]

Which has the solution,

[tex]T = (T_0 + b/c) e^{-cs} - b/c \,\,\,\,\,[/tex] : valid while T>0BTW. Have you done ODE's in maths yet?
 
yeah I've done one or two. I needed to solve it for s(max). The solution to the problem is how far the car travels not the kinetic energy of the vehicle.

from
dT/ds = b+cT

we get

dt/(b+cT) = ds

for

(b+cT)=x, dx = cT

therefore

T = 1/c(dx)

we now get

1/c (dx/x)=ds

integrating

integral b+cT0 - b(1/c){1/x} =integral 0 - smax{s}

we get

smax = 1/c{ln(b)-ln(b+cT0}

The constants are negative.

is there a way ogf using equation editor

I just need to know if I can get

dT = F(net)ds from the variable force if you can help
 

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