How does animal size affect individuals' agility and speed?

Question

This question is a bit of an outgrowth of Can you simply scale up animals?, but it does approach it from a different angle.

"All else equal" (which it of course never is, but we're good at pretending here, so let's pretend), heavier animals tend to be less agile than their smaller counterparts. This to some extent follows from the fact that their larger mass represents a larger inertia, but it would seem like it would be counteracted by the greater force which can be exerted by the (larger) muscles.

I'm basically looking for a way to, within a single species, have a "baseline" size (in terms of weight and possibly height) and a formula where one can plug in the size of the specific individual and the result is some sort of index or factor describing how agile that individual is compared to its peers, to model their relative agility. For the purposes of this question, let's say agility translates directly to how quickly an individual is capable of moving a certain distance from standstill to standstill. (In other words, I have $\Delta{}T = f(d, \dots{})$ where $\Delta{}T$ is time required to move a given distance $d$, and I'm looking for what goes into $f()$.)

I have a feeling that the old workhorse $F=ma$ is going to be at the center of such an equation, but that there is more to it than that. I also recognize that it can't be exact, but for my purposes it doesn't need to be exact; I'm looking more for a rule of thumb than exact mathematics. If it makes it easier, the question can be limited to quadrupedal animals; it would be better if an answer applies at least to both bipedal and quadrupedal animals, but that is not a requirement for a good answer (only a great one).

• Ideally, there is such a formula established in the scientific fields of study. If so, what is it? (I doubt this is the case, but who knows?)
• Alternatively, what factors would go into such a formula for it to be at least semi-accurate? What might it look like?

Note that an answer does not necessarily need to state an explicit formula. A description of the factors that goes into answering this and how they interact can be equally valid.

Answer 1

The factors for an animal are the same as for vehicles in many ways. Agility is essentially a measure of acceleration, and your ability to apply that acceleration in different directions rapidly.

The limiting factors for acceleration as animals grow larger are:

• Muscle Power
• Traction
• Mass
• G Force

But there are also advantages with elements like:

• Stride length
• Leverage
• Energy Storage
• Air resistance

Unfortunately there is no one simple formula you can apply here, each factor can vary greatly. For example a house cat on a carpet where they can dig their claws in is incredibly agile, the same cat trying to make a sharp turn on polished wooden flooring could well slide into the wall.

We can discuss each factor though.

Muscle Power

Larger animals have more power available, the raw strength increases with the square of size (as it's the cross section of the muscle) but total power does increase with the cube of size (as it's the volume of the muscle) so larger animals can end up with plenty of power to work with so long as their tendons and bones can take the strain.

Traction

Claws and hooves are common adaptions to increase traction but you are still limited by the surface area of your feet. Larger animals become increasingly susceptible to problems getting traction and are able to slip and fall more easily unless they can grip something with their claws.

Mass

Mass increases with the cube of size. That mass takes power to accelerate. The positioning and layout of mass should also be considered. Keeping the mass central and being able to adjust it dynamically for example by twisting your body gives you a big boost in maneuverability.

G-Force

Rapid acceleration and deceleration can cause serious injuries, that are easier to handle for smaller animals. Additionally the terminal velocity of smaller animals tends to be lower due to their higher surface area to mass ratio. As a result longer falls are survivable for smaller animals than for larger ones.

Stride length

Larger animals need to take fewer steps to travel the same distance or the same speed than a smaller one. This allows them to grip once and then apply the power smoothly over time rather than constantly re-applying it.

Leverage

Longer limbs give more mechanical advantage to muscles - so long as the muscles can have enough strength to take advantage of it.

Energy Storage

Fleas wind up springs in their legs in order to release it in a single bound - jumping far further than they could without that. Similarly Kangaroos bounce very efficiently due to springs in their legs that store energy when landing then use it to propel the next upwards leap.

Air resistance

Air resistance effects larger creatures less as it is proportional to the square of size whereas mass has increased by the cube of size. A large creature can jump further if it starts the jump at speed X than a smaller creature starting at the same speed just because it is not slowed down by the atmosphere as much.

Conclusion

In general larger creatures do not have to be less agile than smaller ones. Their muscle power is proportional to their mass, they have advantages in terms of leverage and stride length, and they are less restricted by air resistance. However they are also limited by the strength of their bones and how much traction they can develop.

An elephant's muscles and skeleton could be tuned to allow it to run like a Cheetah, however it would snap every tendon and bone in its body and when it tried to accelerate it would throw more dirt backwards than it would go forwards. Rhinos though have been measured running at 55km/h with a surprising ability to turn and maneuver while doing so.

So for a certain range of sizes there is roughly constant speed and turning radius no matter how larger a creature grows. Once the limits in strength of materials and the limits of traction from the ground starts to be reached then the available maneuverability starts to decrease. As maneuverability decreases then increased speeds become more dangerous and less useful so top speeds also tend to decrease as other methods of defense take priority.