Let’s break this down into parts:

send a spaceship into space

Assuming we’re launching from Earth’s surface, we will need to: 1) get safely away from the ground (or else we’ll crash first), and 2) achieve an orbital velocity of at least 11 km/s, which is needed to escape the influence of Earth’s gravitational pull. If we launch from the equator and launch towards the east, we get the free benefit of the equatorial velocity, which is about 0.44 km/h, so that reduces our required speed to “only” 10.56 km/h.

huge slingshot

The thing with all machines that yeet an object into the air is that they’re all subject to the trajectory calculations, again due to that pesky gravity thing. As much as we’d like things to be as easy as “point and shoot” in a straight line out into space, the downward force of gravity means we must aim upward to compensate.

Normally when one thinks of trajectory, it is to aim an artillery piece in such a way that it’ll land upon a target in the distance, but typically at about the same altitude as where it was launched from. But if the artillery piece is perched upon a hill aiming down into a valley, then a smaller angle correction must be made because it would hit farther than intended. When aiming at a target located higher than the gun, the correction would be a slightly larger angle.

In this case, to aim into space – assuming we mean something near the Karman line at 100,000 km above MSL – that’s a substantial height and we’ll need to aim the slingshot with a substantial vertical component. The exact angle will depend on what horizontal component (is velocity) we need, which was discussed earlier.

how much rubber band

The relationship between the necessary vertical component (to overcome gravity) and the horizontal component (to reach escape velocity, which is caused by gravity) can be drawn as two orthogonal vectors, with the rubber band having to provide the angled thrust equal to the sum of those two vectors.

We’ve ignored air resistance, but with this simple relationship, it’s clear that we can use basic trigonometry and the Pythagorean theorem to find that the rubber band vector is the sum of the square of those two vector magnitudes. Easy!

The only problem is that, on its own, the square of some 10.5 km/s is a huge number. Even 10.5 km/s is a huge number, already exceeding the speed of sound (0.343 km/s) many times over. I vaguely recall a rule somewhere that elastic deformation cannot exceed the speed of sound, for reasons having to do with shockwave propagation or something like that.

But I think it’s all fairly intuitive that for a rubber band slingshot to accelerate an object, it too must be in contact with said object while accelerating. And since a rubber band probably can’t reach the speed of sound (and remain intact), then it also cannot accelerate another object to the speed of sound. To then ask for the slingshot to accelerate to 30x the speed of sound would be asking too much.

For this reason, I don’t think the rubber band slingshot to space will work, at least not for a typical linear slingshot. If you do something that rotates and builds velocity that way, then it becomes feasible.

This feels like one of those xkcd questions.

Without involving maths, its fair to say it wouldn’t be unlike having it launched by an explosion.

The closest thing I can recall related to this concept would be Project HARP.

Whatever you’re launching would need to be capable of absorbing the unreal amount of G force and survive the resistance of the atmosphere. As you’ll be required to launch the spaceship at escape velocity with a delta of near 0.

I don’t believe rubber has the material strengths necessary to achieve that launch while still being able to store it on the face of the planet. So, imagine a ball of elastics the size of the sun and maybe there’s a chance because of the slow rate of acceleration achievable by not being on the earth any more.