Need help understanding continuously applied physical quantities like force Given a 1 kilogram object at rest which then begins to free fall for 100 meters, and the only acting force is gravity...

*

*Is it even possible to calculate the total force applied to the object due to gravity throughout its 100 meter descent since the force is being applied continuously (at infinitely small time intervals)?


*If it's impossible, does this relate to all resultant graphs of analog measuring systems (like a seismometer) where there are essentially infinite samples (output values) over any given time period.  You can't simply add up infinite samples and there is no discrete sample time step.  Does trying to find a total just simply not make sense for these types of things?


*Based on the previous 2 ideas, what does the force part of Work = Force x Distance actually mean? I was under the previous assumption that it was total net force applied to a system over a distance but now I'm thinking it is actually the average net force applied over a distance on the object which work is being done upon (given that the force is the cause of displacement).
Any insight is greatly appreciated, thanks.
 A: It seems as if you think that the force "accumulates" over time in the same way that an object might "accumulate" distance over time as it's moving. This is a misconception about the nature of force. The net force on an object at a particular moment in time determines the acceleration of the object at that moment in time.  That's it.  The object doesn't accumulate force, and it certainly doesn't "carry that force with it" after the force has stopped acting.  A force is not a property of an object.  It's something that's done to the object.
However, the force acting continuously over time has a cumulative effect on the object. Here, you want to think about the work done an object, which can cause the object to "accumulate" energy (in the case where the work is positive).  If the a force $\vec{F}$ is acting continuously, then imagine breaking the trajectory of the object up into very many small paths.  If each chunk is small enough, the force is approximately constant on that chunk, and you can compute the work done on that chunk as
$$
W_{\textrm{by }F} = \vec{F}\cdot\Delta \vec{r}\,,
$$
where $\Delta \vec{r}$ is the displacement of the particle during this chunk of the trajectory.  For convenience, let's imagine the object is moving in one dimension, say the $x$-direction, and that the force is also applied in this direction. Then,
$$
W_{\textrm{by }F} = F_x\Delta x\,.
$$
Now, to get the cumulative effect of the force acting on the object over this entire trajectory, you just add up all the works done on each chunk, i.e.,
$$
W_{\textrm{by }F} = \sum_{\textrm{chunk}}F_{x,\textrm{on chunk}}\Delta x_{\textrm{chunk}}\,.
$$
In the limit of more and more smaller and smaller chunks, this becomes an integral
$$
W_{\textrm{by }F} = \int_{x_i}^{x_f}F_{x}dx\,.
$$
This work done then determines the change in energy of the system cause by the cumulative action of the force.
Now, this work can be interpreted graphically as the area under the $F_x$-vs-$x$ graph, but that's a sort of abstract representation of the motion, because it doesn't take into account time (and therefore the rate of motion along the trajectory).  As mentioned in another answer, you can make the graph of the $F_x$-vs-$t$, but the area under that graph is the impulse, which represents the cumulative effect of the force acting over that time in the sense that the impulse determines the change in momentum of the object (rather than the change in energy).

Finally, you can re-write the work done above as
$$
W_{\textrm{by }F} = (x_f-x_i)\frac{1}{x_f-x_i}\int_{x_i}^{x_f}F_{x}dx
= \Delta x \vec{F}_{\textrm{avg}}\,,
$$
where $\Delta x$ is the net displacement of the object, and $\vec{F}_{\textrm{avg}}$ is the spatial average of the force, averaged over its trajectory.  This quantity is pretty abstract, and it's unclear to me whether it's a useful quantity (conceptually).  It is common to define the time-average of the force in the context of collisions, where you are computing impulses and changes in momentum, but I don't think the spatial average is a very useful notion.
A: It is possible to measure the average force over a time. The expression "total force" though makes no sense. It is not like averaging a set of numbers where you find their total and their count, then divide these two. If we were finding the average force from a Force-time graph we would find the area under the graph and divide that by the total time. The area though is not a total force as it is equivalent to a force multiplied by a time. This area is called the impulse of the force.
Work can be found from a force-displacement graph - the graph shows the force on the object at each position during its fall. The work is found by calculating the area of the force-displacement graph. If the force is constant the formula $W=F\times d$ works. A falling object usually is treated as having a constant force on it.
There are many quantities which can only be measured as averages. For example velocity is measured by measuring the distance traveled in a measured time interval. If the time is short enough (very short) the velocity is taken to be near enough to the velocity at every instant during that time interval. If you had a series of adjacent measurements and you wanted to find an average velocity for them all, you could multiply velocity by time for each interval (giving displacement), then add them all and divide by the total time. Just adding the velocities would not give you anything useful unless you knew all the times were the same, and even then it would not refer to any real "total velocity" of anything.
