Spring and/or bungee cord behind vehicle?

[ScienceNostalgia101]

Disclaimer: I do NOT presently intend to actually attempt this myself, NOR would any future attempt at this myself hinge exclusively or even primarily on the direction of this thread. Obviously if I were to ever become a stuntman in the future I would consult with actual experts. For now, this is intended primarily out of curiosity. I do not recommend this for anyone without consultation with actual experts as even to a layman there are a bunch of obvious issues (debris on the road, tripping up and falling face flat on a moving road) that could go wrong.

 

Suppose someone wearing roller blades, or riding a skateboard or bicycle, or whatever else (presume an abundance of safety gear; and perhaps some springs between their multiple suits of armor) attached a spring, or a bungee cord, or a spring attached to a bungee cord, so as to connect themselves or their bicycle to a motor vehicle like a car. Whether for the thrill of it, or as a semi-convenient means to transport their bicycle for lack of room in the car, or both.

 

I presume the non-motorized vehicle would get infinitesimally closer and closer in both acceleration and velocity to that of the motorized vehicle, as there is only so much a spring or bungee cord can stretch. But I'm kind of stuck on how to figure out what the acceleration would be as the vehicle begins to move. Obviously there would be a major gap in both acceleration and velocity as the bungee cord and/or spring stretched to accommodate the fact that the vehicle in front is driving the motion and the non-motorized vehicle is just acting as a load. How much of a gap will that be?

 

All I have right now is that the only force acting forward on the non motorized vehicle is F=-kx; and that the only one acting backwards on it is friction. I have no idea how to figure out what the next step is. I presume from the nature of the question that it's plentifully obvious this is neither a homework question nor a lesson planning question.

[Chopstick]

Simplifications: I think you can ignore friction at first, since you're talking about nice rolling things. A spring on the end of a rod is easier to model than bungee, since that confines stretching to approximately a single point. The starting state is with the spring at its' resting point, everything else is rigid. And the ending state, after the acceleration is done, is a steady state where bike & car have the same velocity, with 0 acceleration.  Oh, also assume the car has uniform acceleration.  For the math, I'm using arbitray time dimensions (ticks; p[resumably significantly less than 1 second per tick), but that should be okay up to a scaling factor. And PS, car is much more massive than bike, bike is much more massive than spring

Exact answer does require chemical or material knowledge to determine k

Since you want to know about the dynamic situation between the steady states, I would assume a numerical/simulation in discrete time is necessary.

Tick 0: everything has v=0, displacement of spring is 0, the car has acceleration a, everything else has acceleration 0.

tick 1: car has velocity v, car has traveled distance v/2. v=at, x=0.5 at^2.  The string is now stretched to length y = 0 + x. F=kx gives you the force on the spring. Assuming no kind of damping factor, the string-stretching force applies to spring & bike. F=m'a' gives you an acceleration for the bike (primes are values for the bike, unprimed is car)

tick 2: car has velocity 2v, bike has velocity v', car is at position x=2a, bike is position x'=0.5 a'. Spring length is X - X'. Calculate force on spring, repeat calculation for spring force F, bike accelerationj and velocity a' and v'

repeat calculations every tick, untiil car velocity and bike velocity are however close you want them to be.

That's my rough draft of a procedure at least.