Find the best T(t) from a differential eq. and conditions

In summary, an electric vehicle of mass ##m## moves along the ##0-x## axis according to the law:m\frac{\mathrm{d} v(t)}{\mathrm{d} t} = T(t)-\mu mg-kv(t)^{2} (ma = Thrust - Friction - Drag)It is known that:P(t) = T(t)v(t) (Power = Thrust * Speed)Find the thrust ##T(t)## in such a way that the vehicle be able to reach a destination as far from origin as possible with a given amount of energy, ##E##, stored in
  • #1
simplex1
50
1
An electric vehicle of mass ##m## moves along the ##0-x## axis according to the law:

[itex]m\frac{\mathrm{d} v(t)}{\mathrm{d} t} = T(t)-\mu mg-kv(t)^{2}[/itex] (ma = Thrust - Friction - Drag)

It is known that:

[itex]P(t) = T(t)v(t)[/itex] (Power = Thrust * Speed)

Find the thrust ##T(t)## in such a way that the vehicle be able to reach a destination as far from origin as possible with a given amount of energy, ##E##, stored in its batteries, in a time [itex]\tau < \tau_{0}[/itex].

[itex]E = \int_{0}^{\tau}P(t)dt[/itex]

The quantity that has to be maximized is the distance, ##d##, covered:

[itex]d = \int_{0}^{\tau}v(t)dt[/itex]

##E, m, \tau_{0}, g = 9.81 m/s^2, \mu, k##, the last two being the friction and drag coefficients, are all parameters with given values.

I have no idea how to solve such a problem. Some suggestions about finding ##T(t)##, in Mathcad for instance, would be welcomed.
 
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  • #2
Relating a change in energy to a distance ... that would be work-energy theorem.
 
  • #3
This is a tough math problem. All the necessary equations have been already written.
 
  • #4
Presumably you can also use physics.
Anyway, you wanted to know how to start... you start by playing around with the relationships and reviewing your knowedge so far like in course notes.
 
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  • #5
I will reformulate the problem in a pure math language:

Find ##x(t), y(t)## that maximize the quantity ##\int_{0}^{\tau}x(t)dt##.

It is known that:

##\frac{\mathrm{d} x(t)}{\mathrm{d} t} = a\frac{y(t)}{x(t)}-bx(t)^{2}-c##

and

##d = \int_{0}^{\tau}y(t)dt##

where ##a, b, c, d, \tau## are given parameters available as numerical values.
 
  • #6
... you start by playing around with the relationships and reviewing your knowledge so far like in course notes.

Unless you make some attempt at the problem, it will be difficult to help you.
Experiment with the relationships, try stuff out, gain an understanding of what's going on, something should come to you.
Not everything is a matter of knowing what to do from the beginning.

i.e. One of the keys will be to realize what you are maximizing with respect to.
 
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  • #7
Sorry but if you do not know to solve such a problem do not answer because you discourage other people that may have the solution by discrediting me with your comments suggesting that I do not know what I want when in reality the problem is clearly formulated.
 
  • #8
This is an interesting problem and I would like to see the solution.
to get thrust as a f(t) does this approach help any?
Dimensionally, Distance = [(Power x Time) x (Distance / Time)] / Power
That is: Distance = Energy (Velocity/Power)
Thrust = Power/Velocity
Distance = Energy/Thrust

Thrust = E given / ∫ V dt
Now to work that Thrust (t) into the differential equation.
 

1. How do you find the best T(t) from a differential equation and conditions?

In order to find the best T(t), you will need to solve the given differential equation and apply the given initial conditions. This will result in a particular solution for T(t) that satisfies both the equation and the conditions.

2. What is the purpose of finding the best T(t) from a differential equation and conditions?

The purpose of finding the best T(t) is to determine a function that represents the behavior of a system over time. This can be used to make predictions and analyze the system's behavior under different conditions.

3. Can the best T(t) be found using numerical methods?

Yes, the best T(t) can be found using numerical methods such as Euler's method or Runge-Kutta methods. These methods involve approximating the solution to the differential equation at different points in time and can be used to find an accurate solution for T(t).

4. What are the initial conditions used for in finding the best T(t)?

The initial conditions represent the starting point of the system and are necessary for finding a specific solution for T(t). Without these conditions, the solution would be a general solution that could represent any system with the given differential equation.

5. Is it possible to find multiple solutions for T(t) from a single differential equation and set of conditions?

Yes, it is possible to find multiple solutions for T(t) from a single differential equation and set of conditions. This can occur when the equation has multiple possible solutions or when the conditions are not specific enough to determine a unique solution.

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