Good news: I found a study that uses musculoskeletal simulation to predict the jumping ability of a theropod!
P J Bishop, A Falisse, F De Groote, J R Hutchinson, Predictive Simulations of Musculoskeletal Function and Jumping Performance in a Generalized Bird, Integrative Organismal Biology, Volume 3, Issue 1, 2021, obab006,
doi.org/10.1093/iob/obab006Bad news: The theropod in question is Eudromia elegans, the elegant crested tinamou, this thing:
No hate on the bird, it's quite a pretty bird and tinamous are cool, being volant ratites, and I guess this is good news if you're interested in their jumping ability, though I doubt most people in this thread were...
Still, there's a lot of information there so give it a read through the link above. One passage that I found interesting though was this:
Apparently increasing the power of the muscles that actuate more distal limb elements had a greater effect than increasing the power of muscles that actuate more proximal element, suggesting greater importance of the former, which calls into question the leaping ability of short-footed Eudromaeosars even though most modern interpretations of their hunting strategy requires leaping. As for why this is the case, I'm guessing it's something to do with the fulcrum that each limb element hinges on as it rotates during the leap. The three main limb segments would act as levers as they rotate (flexing metatarsals, extending tibia, flexing femur). The metatarsals' fulcrum is the ground itself, while the tibia's would be the ankle joint (one step removed) and the femur's would be knee joint (two steps removed). Hence, more of the metatarsal's energy is going into the torque that propels the leap while some of the tibia's and femur's will be wasted bending the limb elements beneath it. This could explain why distal elements seem to contribute more to the leap, at least in the tinamou. And since the metatarsals are the most effective levers, it would be advantageous to lengthen these lever arms over more proximal elements. This is also congruent with one of my arguments for why large ungulates would be better jumpers than similar sized Eudromaeosaurs. They have long distal limb bones (in addition to all the other benefits for leaping that being a quadrupedal mammal provides).
(Also disclaimer, this post is gonna include a moderate amount of physics which I am probably not qualified to talk about so take everything with an alkaline lake's worth of salt)
However, to play devil's advocate against myself, I don't think this necessarily means that things like Utahraptor and Achillobator were worse jumpers than similar sized theropods. I suspect that the benefit of long metatarsals diminish with increasing mass and would certainly be much less for a 300 kg raptor than a 1 kg bird. Of course, theoretically, a longer lever produces more torque, leading higher jump velocities. However, longer foot bones also lengthen the moment arm from the animal's weight to the ankle joints (as well as to the proximal ends of the metatarsals) when the feet are extended before and during a jump, which exerts more force on these parts, which not only then necessitates more force to flex them and push off but also poses a risk damaging these elements which are braced against nothing but hard, unyielding ground. In this case, shorter metatarsals are advantageous in that they would require less muscle strength to flex and would be more durable. Consequently, they can extend to a greater degree and therefore flex through a greater arc, allowing more time to apply force to the ground. You can kind of see this with how leaping felids will move their short metatarsals from almost perpendicular to the tibia to full flexion, compared to leaping antelopes which move their long metatarsals through much shallower arcs (We do not talk about cheetahs and how each of their stride is effectively a 7m horizontal leap...). Granted, idk whether this is the antelopes compensating for the long moment arms or cats making the best of their comparatively stunted equipment.
All that said, I'm not suggesting that the short metatarsals of Eudromaeosaurs evolved as an adaptation for leaping, simply that it may not be as clear cut of a disadvantage as the authors or myself in previous posts may have led on. In things like Bambiraptor or Velociraptor, it would be a disadvantage for leaping compared to longer legged theropods of similar sizes (but they're so light weight anyways that it would hardly matter). Deinonychus probably wasn't a champion high jumper by any chance but it's not like Tenontosaurus was particularly far of the ground. Ig my point is more so that Utahraptor and Achillobator wouldn't necessarily have faired worse in this regards than similar sized, stilt-legged theropods, which could only have swung their metatarsals through shallow angles, fine for sprinting, less so for leaping.
Now, with all that being said, a short lever is still suboptimal for obvious reasons. So how would a large Eudromaeosaur, if it had to leap high and far, compensate? Well, one way is just straight up buffing up the legs, what I like to call the crocodile approach (Cuz, to be a bit reductionist, crocodile skulls don't seem to be as built for high bite forces as some other animals with their elongated, tapered snouts but they didn't give a **** about leverage and just slapped on more jaw muscles til they can explode zebra faces.). Obviously both the giant Eudromaeosaurs did this, I mean just look at their femurs and Achillobator's pelvis.
IIRC John Ostrom also noted specialized femoral retractors attached to the hip of Deinonychus which he speculated could have helped with swinging the sickle claw in a disembowling kick, but I don't see why it wouldn't be beneficial for kicking off the ground. Quick tangent on that note, could the reorganization of muscle mass away from the M. caudofemoralis and more towards the hip be beneficial for leaping? Shorter muscle contractions with less of a horizontal component could be beneficial for a stronger vertical leap. And also more mass being concentrated towards the leg instead of away from it may help too.
Another way for something like Achillobator to compensate for its plesiomorphic short metatarsals is to lengthen the other lever arms. This seems counterintuitive for something that's often called out for being stocky and short legged, due to its tibia being shorter than its femur (unusual amongst Dromaeosaurs or small-medium sized Coelurosaurs in general). However, mildly hot take, I don't think Achillobator had a short tibia or metatarsals (by Eudromaeosaur standards), I think it had a long femur. Recently, I went on Theropod Database and did some quick maths. Achillobator's tibia was 57.3% longer than Deinonychus (AMNH 3015). Metatarsal III was 55.2% longer. Radius was 51.2% longer. Coracoid was 56.1% longer (Using YPM 5236 for Deinonychus). Maxilla was 66% longer (Using YPM 5232 for Deinonychus). Now, femur was *drumroll please*
77.8% longer. In my unprofessional opinion, it intuitively seems more reasonable to assume that the Achillobator holotype generally had limb bones around 50-60% larger than an AMNH 3015 sized Deinonychus but a femur 80% larger, with jaws somewhere in between, than that it had a regular femur and relatively diminutive everything else. Also, if we assume similar dimensions to Utahraptor, scaling down the 5.1m individual with a 60 cm femur to the Achillobator holotype's 50 cm femur gives 4.25m which is 41.7% longer than a 3m AMNH 3015. 50% using a more generous 4.5m. So basically, Achillobator may have had long thighs, not short shins, which then could act as longer levers for a stronger leap. And the femurs were very robust and well muscled so the drawbacks mentioned above for lengthened metatarsals are less relevant. Besides, IF Achillobator performed RPR, a longer femur gives it better control over the tibia and metatarsals, like a lower brachial index in large prey hunting big cats.
