# Physics Application - Displacement, Velocity, Acceleration, and Jerk

Differentiation is highly used in physics. We will talk about elementary examples connected to physics with regards to displacement/position, velocity, acceleration, and jerk. We will not touch on what these terms necessarily mean in detail with regards to physics however.

# Displacement (Position)

Displacement/position refers to the location of some object from the origin. Suppose that we have a function $s(t) = 3 \sin t$ that describes the displacement/position of a wave from the shore of a beach at time $t$. If we want to know what the position the wave from the shore is, $s(t)$, that the tide is at time $t = 3$, we can simply plug this value into the known function get $s(3) = 3 \sin (3) \approx 0.423...$. This is nothing groundbreaking of course, however, we can get more information by $s(t)$ by taking derivatives of $s(t)$.

# Velocity

Velocity is described as the rate of change of displacement/position and can also be represented by a function, usually $v(t)$. Since $s(t)$ represents displacement, then $v(t) = s'(t)$. If we know a displacement function $s$, we can differentiate $s$ to get a velocity function $v$

Back to our shore example, if we have $s(t) = 3 \sin x$ and want to find a velocity function $v(t)$, then we differentiate to get $v(t) = s'(t) = 3 \cos x$. If we want to know the velocity of the tide at time $t = 3$, we thus plug this value into $v$ to get $v(3) = 3 \cos (3) \approx -2.96$. Therefore, the tide is receding at a velocity of $v = -2.96$.

# Acceleration

Acceleration is acknowledged as the change in velocity with respect to time. Hence, we can say that some acceleration function $a(t)$ is equal to the derivative of our velocity function $v(t)$ which is equal to the second derivative of our displacement/position function $s(t)$, and thus we build the connection that $a(t) = v'(t) = s''(t)$.

Back to our tide example, since $v(t) = 3 \cos x$, if we differentiate this again we obtain $a(t) = -3 \sin x$, and computing how the tide is accelerating at time $t = 3$ is as simple as plugging into $a$ again, that is $a(3) = -3 \sin (3) \approx -0.423...$.

# Jerk

Jerk is described as the rate of change of acceleration with respect to time, $j(t)$. Often omitted as an early example of differentiation, we get the relation that $j(t) = a'(t) = v''(t) = s'''(t)$.

For our tide example, we know that $a(t) = -3 \sin x$, so $j(t) = a'(t) = -3 \cos x$, and at time $t = 3$, $j(3) = -3 \cos (3) \approx 2.96...$.

The following graph illustrates the functions of displacement, ${\color{Red} s}$, velocity, ${\color{Orange} v}$, acceleration, ${\color{Green} a}$, and jerk, ${\color{Blue} j}$, used in our wave/tide example plotted on the same grid.