The velocity of a rocket bike depends on the type of rocket attached to the bike. More specifically, It depends on the Delta-V that the rocket can impart on the mount that is connected to the bike. Delta V is the effective exhaust velocity c of the rocket times the natural log of the initial mass over the final mass.
The initial mass will be the mass of the biker, the bike, the rocket mount, and the rocket itself. The final mass will be everything except for the mass of the propellant used in the rocket.
The change in velocity of the bike will be negligible if the mass of the propellant used is not a significant percentage of the bike’s mass.
The effective exhaust velocity can be found by multiplying the gravity at sea level by the rocket’s specific impulse.
If you take a G class rocket which is the largest rocket motor that a person can buy without needing a certification, then you’ll roughly have 160 Newton seconds of total impulse.
To go from total impulse to specific impulse, divide the total impulse by the weight of the rocket motor.
This will lead to an effective exhaust velocity of 1406 m/s.
Now you’ll need the weight before and after the rocket has spent its propellant. Let’s say that bike has a mass of 18 lbs with 175 lbs. person driving the bike. The 130 grams rocket motor and rocket mount weighs about 87671 grams. Let’s consider that mass of everything after the rocket has burnt all of the propellant is 87110 grams.
Given that the average burn time for one of these rocket motors is 3 seconds, the acceleration that the rocket will experience is
The total distance traveled, negating friction will be 2.6 meters.