American said:
How's about a really, really, REALLY big rubber band?
How practical is this idea really?
The elastic potential energy in the stretched band is given of course by
(1/2)*kdx^2, ….1
where k is the constant of proportionality in the simple harmonic approximation, or ‘spring constant’, and dx is the change in length.
Now in the absence of non-conservative forces the potential energy would be fully converted into kinetic energy, which in turn is given by (1/2)*mv^2, where m is the mass of the projectile and v is its final velocity. We can then express the extension as
dx = v*sqrt(m/k), ….2
Now what would dx be in practice? Let us assume a spring constant of 100 N/m, a projectile mass of 1000 kg, and a required final velocity of 10 km/s (basically between orbital and escape velocity). Substituting these values into Eq. 2 gives an extension of
dx ~ 31.6 km.
If we further assume the band can stretch about 5 times its unstretched length before breaking we would have to use a band of nearly 8 km. Also note of course that this the
minimum possible length, corresponding to the assumed values. I’m sure you’ll agree that this is hardly practical. Now I leave it up to you, American, to calculate the maximum
acceleration imparted to our projectile and what the implications would be for any human being traveling in it….