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A cylinder is slung beneath the chassis facing backwards. The piston is attached to a rod, at the other end of which is a spiked plate designed to dig into and grip the road.
Fuel is admitted to the cylinder and ignited, forcing the piston and rod and road backwards, and the vehicle therefore forwards.
piston rod is attached to a complicated linkage assembly which retrieves it and inserts it once again into the cylinder, but facing forwards this time. The forward momentum of the vehicle allows the road to push the plate and therefore the rod and therefore the piston up the cylinder, compressing the newly-admitted fuel. Just after passing top dead centre, the fuel is ignited.
Steam Horse locomotive
Brunton's Mechanical Traveller [8th of 7, Apr 16 2018]
MEDHURST'S GUNPOWDER ENGINE
Similar [8th of 7, Apr 16 2018]
Existing Variant Design
Now take that existing design, turn it inside out so the piston-rods hit the ground like you describe, and allow the overall engine to rotate. [Vernon, Apr 16 2018]
[MaxwellBuchanan, Apr 17 2018]
Precursor to tracked vehicles [8th of 7, Apr 18 2018]
Early aviation engine [8th of 7, Apr 18 2018]
Single Cylinder Pneumatic Vehicle
[bigsleep, Apr 19 2018]
||Very similar to Brunton's Mechanical Traveller ...
||Does the very complicated assembly account for the
inertia of the piston and rod when the moving car
tries to pick it up again?
||A pedrail wheel <link> combined with a true rotary 2-stroke IC engine <link> would be interesting. Particularly from the point of view of speed control, because a crankcase-scavenged rotary can't be throttled, only run "hit-and-miss".