Can the propulsion system be used to then propel the projectile?
IE: Can the 1 m initial movement be used like a discus hurler's start?
Does the projectile need to accomplish anything other than contact, so it could be a the a fore mentioned soft air pellet, or does it have to knock something over/put a hole in a target, etc? How large is the target/how important is accuracy?
Does the initial 1 m have to be in a straight line, or can the path be curved/intersect a point at the 1 m mark, etc?
You mentioned it can only move by elastic or gravitation forces, but, not if the projectile firing was also limited to that. The spring of a spring action airsoft gun should count as elastic I would think.
Can the projectile itself have a spring, or compressed air or water, etc...and the firing mechanism simply holds it in, compressed until firing?
What is allowed as far as initiation of the progress forward or firing? Are you allowed to do one action to start it towards the 1 m point, and then do other actions to hit the target, or is it supposed to start towards the target and fire by itself?
As mentioned, on sand, you want it low, and light, with flotation type traction and good ground clearance so it doesn't bog down...and tank tread type tread, or at least 4wd, rolling ball, etc.
The sand can also complicate aiming, as you might be leaned over or tilted due to the terrain...so when you stop, your pitch/yaw might impact where your shot goes, especially if the target is small, etc.
The power and accuracy required will probably dictate the size and weight of your platform/form factors...as a stronger throw, etc, will require a more secure base....and higher degrees of accuracy, if you are NOT able to "aim" after you stop, etc...will require some sort of compensation for pitch/yaw for your firing mechanism.