Mechanics Problem: Finding Displacement, Velocity, and Acceleration Formulas

[lavalamp]

Homework Statement

Hey all, this isn't actually a homework question, but I guess it's of that type. For some time now I've had this (not entirely realistic) mechanics problem that I keep leaving for a while, and then coming back to. Basically, I'm not getting anywhere so I am asking for some help with it. Ideally I'd like to find formulae for s, v and a in terms of time, and I'd appreciate any help people can offer.

It's an idealised car with power 100 kW, mass 1000 kg, and a seemingly infinite amount of grip. There is also an aerodynamic drag force which I've set at -0.5v^2. The car begins accelerating from rest at t=0 and I'd like to find it's displacement, velocity and acceleration at a given time.

Homework Equations

I can get a formula for acceleration in terms of velocity relatively easily:

P = Tv
T = P/v

F = ma
T + Fd = ma
P/v - 0.5v² = ma
100000/v - 0.5v² = 1000a
200000/v - v² = 2000a

a = (200000/v - v²)/2000

$$a = \frac{100}{v} - \frac{v^{2}}{2000}$$

The question is, what comes next?

The Attempt at a Solution

Here are two equations I arrive at when I attempt to progress a little further, I'm fairly confident that they are both incorrect:

$$v^{3} = \frac{600000s}{3s + 2000}$$

$$s^{3}\ +\ 3000s^{2}\ +\ hs\ =\ 200000t^{3}\ +\ 6000000*5^{1/3}t^{2}\ +\ 20*5^{2/3}ht$$

c, e, f and h are constants. I actually couldn't eliminate h.

Here's how I came up with the first equation:

a = 100/v - v²/2000
2000a = 200000/v - v²
2000va = 200000 - v³

$$2000v\ \frac{dv}{ds}\frac{ds}{dt} = 200000 - v^{3}$$

$$2000v^{2}\ \frac{dv}{ds} = 200000 - v^{3}$$

$$\int 2000v^{2}\ dv = \int 200000 - v^{3}\ ds$$

$$\frac{2000v^{3}}{3} = 200000s - v^{3}s + c$$

$$v^{3}s + \frac{2000v^{3}}{3} = 200000s + c$$

3v³s + 2000v³ = 600000s + c

v³(3s + 2000) = 600000s + c

$$v^{3} = \frac{600000s + c}{3s + 2000}$$

When t=0, s=0 and v=0, therefore c=0.

$$v^{3} = \frac{600000s}{3s + 2000}$$

And the second one:

Calculating the maximum velocity of the car (used later):

P = Tv
P = 0.5v² * v
P = 0.5v³
200000 = v³
v = $$\sqrt[3]{200000}$$ ~= 58.48 m/s

a = 100/v - v²/2000
2000a = 200000/v - v²
2000va = 200000 - v³

2000v dv/dt = 200000 - v³

$$\int 2000v\ dv = \int 200000 - v^{3}\ dt$$

1000v² = 200000t + c - d³s/dt²

$$\int \int 1000v^{2}\ d^{2}t = \int \int 200000t + c \ d^{2}t - \int \int \int d^{3}s$$

$$\int \int 1000\ \frac{d^{2}s}{dt^{2}}\ d^{2}t = 100000t^{3}/3 + ct^{2} + et + f - s^{3}/6$$

$$\int \int 1000\ d^{2}s = 100000t^{3}/3 + ct^{2} + et + f - s^{3}/6$$

500s² + hs = 100000t³/3 + ct² + et + f - s³/6

s³/6 + 500s² + hs = 100000t³/3 + ct² + et + f

s³ + 3000s² + hs = 200000t³ + ct² + et + f

When t=0, s=0, therefore it's easy to spot right away that f=0.

When t is very large, v goes to cuberoot(200000) (max velocity calculated earlier), therefore s goes to cuberoot(200000)t. So setting s=cuberoot(200000)t

200000t³ + 3000*200000^2/3t² + 200000^1/3th = 200000t³ + ct² + et

6000000*5^1/3t² + 20*5^2/3th = ct² + et

c = 6000000*5^1/3
e = 20*5^2/3h

s³ + 3000s² + hs = 200000t³ + 6000000*5^1/3t² + 20*5^2/3ht

 

[Oerg]

Hi,

To get your first equation, the integral is done wrongly. v is treated as a constant in the integral when it is not.

The equation you are trying to solve is a second order non-linear differential equation. a is a second order differential of s and v is a first order differential of s.

 

[lavalamp]

Thanks for the reply. Yeah, for the integration in the first one I had a bad feeling about that as I was doing it, but I didn't, and don't, know what else to do there.

Do you have any suggestions about how I can um ... not do it wrong? I'm pretty much at the limit of what I know here, which is not very much.

For the second equation I spotted a mistake, so starting from here again:

$$\int \int 1000v^{2}\ d^{2}t = \int \int 200000t + c \ d^{2}t - \int \int \int d^{3}s$$

$$\int \int 1000\ \frac{d^{2}s}{dt^{2}}\ d^{2}t = 100000t^{3}/3 + ct^{2} + et + f - \int \int s\ +\ g\ d^{2}s$$

$$\int \int 1000\ d^{2}s + \int \int s\ +\ g\ d^{2}s = 100000t^{3}/3 + ct^{2} + et + f$$

$$\int \int s + 1000 + g\ d^{2}s = 100000t^{3}/3 + ct^{2} + et + f$$

s³/6 + 500s² + gs² + hs = 100000t³/3 + ct² + et + f

s³ + 3000s² + gs³ + hs = 200000t³ + ct² + et + f

s³ + (3000+g)s² + hs = 200000t³ + ct² + et + f

And again, when t=0, s=0, therefore f=0.

s³ + (3000+g)s² + hs = 200000t³ + ct² + et

Replacing constants:
a = 3000 + g
b = h

s³ + as² + bs = 200000t³ + ct² + et

And the trick I pulled last time, setting s equal to the cuberoot of 200,000 multiplied by t, I don't think is valid. Which means I'm stuck with 4 constants I don't know how to find the values of.

However, I ran four simple simulations of this car and found 4 values for s and t, from which I used simultaneous equations to find:

a = 2229.30
b = -252.85
c = -3673.88
e = -454.72

I'm not sure what the accuracy of these values is, I'd have to guess not too great since I ran a fifth simulation and solved the simultaneous equations again to get somewhat different values:

a = 2229.10
b = -257.97
c = -3731.14
e = -464.78

 

[Oerg]

Well... non-linear odes are hard to solve, I'm not proficient at it either, and scientists often use numerical methods to calculate whatever they need if they can't solve the ode. Maybe you could simplify your problem to a power one dependence of the velocity on the drag force, or you could look up some engineering books to see how they did it.

If you require a numerical solution, look up the Runge-Kutta method.

The first integral is done wrongly for your second equation. integrating v^3 with respect to t is not d3s/dt3. I see that you are attempting to solve the same integral as with the first equation.