Worn or broken motor mounts can do this. I just replaced mine on my 98 Cobra. Ford as some new ones that are supposed to be a bit stronger than the stock ones. They look identical to the ones I removed but the rubber must be stronger.
98 Tan on Laser Red Rag Top Cobra
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Fully built and forged with Mahle pistons, SHM Blower cams, ported heads/nitrated valves
Novi 2000, 52# RC injectors, custom IC, Devilsown alky injection, SSR Intake, Steeda rails and -8 feed lines, 03 tank + Aviator dual pumps
Tuned with SCT Pro Racer Package (and everything else over the years)
Lot's of other stuff!