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| Image from Physics in History account on X |
Edit: The answer!
My guide to basic calculus explains this in more detail and might be worth
reading first, but this is as simple as I can make it without a lot of
diagrams.
The
variable "x" is often used in physics and mechanics for representing
distance from a reference position or displacement. So for instance I
could draw a graph of the distance travelled by a car from a start point
versus time or the displacement of a swinging pendulum from its rest
position. x is a function of time, so often it's expressed as x(t). For a
moving object, we can work out average speed by dividing total distance
travelled over a journey by time taken. (50 mph in the diagram below).
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| Explaining calculation of instantaneous speed. © Eugene Brennan |
However if the object travels at various speeds, a more valid
calculation is to determine the instantaneous speed. This is the rate
of change of distance x versus time t, known as the first derivative of x
with respective to t and expressed as dx/dt. This is the instantaneous
speed of the object (usually the term "velocity" is used in physics
which means speed in a given direction). Average velocity can be worked
out over small distances by calculating the slope, i.e. Δx/Δt as shown
in the diagram above. (I used Δs for small distances). As the distance gets vanishingly small, in the limit,
the instantaneous velocity becomes the slope of a tangent to the x
versus t graph as shown in the diagram I made below. I.e dx/dt.
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| The derivative of distance travelled wrt time at any point is equal to the slope of a tangent at that point and is the instantaneous speed. © Eugene Brennan |
If dx/dt is
now calculated for every point on the distance versus time graph above and a graph drawn of dx/dt for every point t, then the slope of that graph is the
acceleration of the object, the second derivative of x wrt t and
expressed as d²x/dt². If a third graph of d²x/dt² is drawn versus t, i.e. a
graph of acceleration over time, that graph would be just a horizontal
line for an object whose velocity is increasing at a constant rate, i.e.
velocity increasing uniformly so acceleration is constant. The slope of a tangent to the graph
would be zero. If you're in a vehicle
and it's accelerating at a constant rate, you just feel a force at your
back. However if the acceleration is non-uniform, you feel a jerk. In
that case the slope of the acceleration graph is d³x/dt³ , which is the
third derivation of x wrt t and known as "jerk" or a sudden change in
acceleration. So, don't be a jerk!