I'm afraid there's no unambiguous way to take the depth of your horse's
hoofprints in the sand and convert that into a force. There can be a
lot of variation in the response of the sand to a horse's footprint
from place to place, having to do with how wet the sand is, its
compositon and the shape of the grains, and the presence of pebbles or
seaweed (if you're on the beach) or anything else. There's also the
problem that the force a horse's hoof exerts on the ground is a rapidly
varying function of time. At the beginning of the collision, the force
is small; it gets much bigger, and perhaps stays constant for a while
or varies with the horse's position and muscle flexing, and then it
reduces until the hoof lifts off for the next step. So then the
question is if you are interested in the maximum force or the average
force over the time the hoof is in contact with the ground.
The average force might be easier to compute from other arguments.
You can weigh the horse on a suitable scale, divide by four (an
approximation -- the four legs may not evenly distribute the weight on
average), and find out the fraction of the time each hoof stays in
contact with the ground at each gait. (weight/4)/(fraction of time the
hoof is in contact with the ground) should give you the average force
on a hoof while it is in contact with the ground. At many gaits, a
horse may spend some time with no hoof in contact with the ground,
while at others, all but one hoof may be in contact with the ground.
You could use a video camera and count the frames in slow-motion over a
full cycle of the periodic step to determine what fraction of the step
has each hoof in contact with the ground.
Now that's the average force -- the maximum force also depends
strongly on what the ground's made of. If the horse is walking on
concrete, the peak force will be much larger than it is on sand. And
not all steps will have the same force of the hooves on the ground --
say, if the horse jumps or lands after a jump, some of the forces can
get very large. Forces can also be locally very high. If the horse
steps on a rock and the total force which normally is distributed over
the contact area of the hoof gets concentrated at a very small contact
area where the rock touches the hoof, the foces can get large.
You could put scales down with ratchets on their dials to record
the maximum force -- but then you'd have to convince your horse to step
on them in a sensible approximation of the gait you're interested in.
There is also such stuff as "pressure sensitive tape" which you can
attach to the shoes of the horse -- it's read out electronically via an
ohmmeter (the resistance of the tape changes with pressure). You'd need
some electronics to record the pressure as a function of time and find
(published on 10/22/2007)