- #1

lavalamp

- 279

- 1

## 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 - 0.5v

^{2}= ma

100000/v - 0.5v

^{2}= 1000a

200000/v - v

^{2}= 2000a

a = (200000/v - v

^{2})/2000

[tex]a = \frac{100}{v} - \frac{v^{2}}{2000}[/tex]

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:

[tex]v^{3} = \frac{600000s}{3s + 2000}[/tex]

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

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 - v

^{2}/2000

2000a = 200000/v - v

^{2}

2000va = 200000 - v

^{3}

[tex]2000v\ \frac{dv}{ds}\frac{ds}{dt} = 200000 - v^{3}[/tex]

[tex]2000v^{2}\ \frac{dv}{ds} = 200000 - v^{3}[/tex]

[tex]\int 2000v^{2}\ dv = \int 200000 - v^{3}\ ds[/tex]

[tex]\frac{2000v^{3}}{3} = 200000s - v^{3}s + c[/tex]

[tex]v^{3}s + \frac{2000v^{3}}{3} = 200000s + c[/tex]

3v

^{3}s + 2000v

^{3}= 600000s + c

v

^{3}(3s + 2000) = 600000s + c

[tex]v^{3} = \frac{600000s + c}{3s + 2000}[/tex]

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

[tex]v^{3} = \frac{600000s}{3s + 2000}[/tex]

And the second one:

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

P = Tv

P = 0.5v

^{2}* v

P = 0.5v

^{3}

200000 = v

^{3}

v = [tex]\sqrt[3]{200000}[/tex] ~= 58.48 m/s

a = 100/v - v

^{2}/2000

2000a = 200000/v - v

^{2}

2000va = 200000 - v

^{3}

2000v dv/dt = 200000 - v

^{3}

[tex]\int 2000v\ dv = \int 200000 - v^{3}\ dt[/tex]

1000v

^{2}= 200000t + c - d

^{3}s/dt

^{2}

[tex]\int \int 1000v^{2}\ d^{2}t = \int \int 200000t + c \ d^{2}t - \int \int \int d^{3}s[/tex]

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

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

500s

^{2}+ hs = 100000t

^{3}/3 + ct

^{2}+ et + f - s

^{3}/6

s

^{3}/6 + 500s

^{2}+ hs = 100000t

^{3}/3 + ct

^{2}+ et + f

s

^{3}+ 3000s

^{2}+ hs = 200000t

^{3}+ ct

^{2}+ 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

200000t

^{3}+ 3000*200000

^{2/3}t

^{2}+ 200000

^{1/3}th = 200000t

^{3}+ ct

^{2}+ et

6000000*5

^{1/3}t

^{2}+ 20*5

^{2/3}th = ct

^{2}+ et

c = 6000000*5

^{1/3}

e = 20*5

^{2/3}h

s

^{3}+ 3000s

^{2}+ hs = 200000t

^{3}+ 6000000*5

^{1/3}t

^{2}+ 20*5

^{2/3}ht