The graphs in the accompanying figure show the position s ,...

The graphs in the accompanying figure show the position $s$ , the
velocity $v=d s / d t,$ and the acceleration $a=d^{2} s / d t^{2}$ of a body
moving along a coordinate line as functions of time $t .$ Which graph
is which? Give reasons for your answers.

Key Concepts

Position Function

The position function describes the location of a moving body along its path as a function of time. Its graph shows how the object's position changes over time, and notable features like slopes, turning points, and steady regions can provide insight into the motion dynamics.

Velocity Function

The velocity function is the first derivative of the position with respect to time. Its graph represents the rate of change of position and is characterized by its correspondence to the slope of the position graph. Positive values indicate movement in one direction, while negative values indicate the opposite direction, and zeros correspond to moments when the object changes direction or is momentarily at rest.

Acceleration Function

The acceleration function is the derivative of the velocity function or the second derivative of the position function with respect to time. It describes how the velocity changes over time, and its graph reflects rapid changes, constant acceleration, or deceleration. Points where the acceleration graph crosses the time axis indicate changes in the acceleration’s sign, affecting the object's speeding up or slowing down.

Graphical Analysis in Kinematics

Graphical analysis in kinematics involves interpreting the relationships between the position, velocity, and acceleration graphs by understanding their derivative connections. By examining slopes, curvature, and intersections with the axes, one can distinguish between the graphs of position, velocity, and acceleration and reason about the nature of the motion.

Transcript

00:01 We're told that this graph here represents the position, velocity, and acceleration of somebody.

00:09 And we want to determine which graph is which, and give reasons for our answers.

00:18 So let's just first make some observations about each of the graphs that we have here.

00:25 So a, well, a is always increasing.

00:30 So that means the derivative of a, so a prime, is always going to be bigger than zero.

00:39 And maybe i'll write a prime of t.

00:42 So a prime of t is always bigger than zero.

00:46 And we also know that a is first less than zero.

00:55 So a of t is less than zero.

01:00 Then a of t is bigger than zero at some point.

01:05 Right here.

01:07 And so we want to look at the slope or the derivative as well as our values for it.

01:16 Because remember, the values of it, if these are derivatives, will then tell us whether the function is increasing or decreasing.

01:28 So we're just going to write all these out for each of them.

01:31 And between all this, we should be able to piece together, which is the position.

01:37 Velocity, and acceleration.

01:39 Now for b, well, the derivative of b is going to be negative until we hit this minimum here.

01:50 So first, a prime, or not a prime, but b prime, b prime of t is going to be less than zero.

02:03 Then b prime of t is bigger than zero, since then it's in increasing after our minimum.

02:15 And we also have that b is always less than zero.

02:19 So b of t is always less than zero...

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