Reading Velocity from Position Graphs

9 Reading Velocity from Position Graphs

In the previous chapter we learned velocity is the derivative of position, $\vec v \equiv d\vec r/dt$ . Let’s do one quick problem to make sure we’re good here.

Exercise 9.1: Calculating Velocities

A particle moves along the trajectory $\vec r(t) = [v_0t+qt^3]\hat x$ . Find $\vec v(t)$ .

You should also tell me whether the object is moving in a line or not, and whether it is speeding up or slowing down (you can assume $v_0$ and $q$ are both positive).

In your calculus class, you probably heard that the derivative of a function tells you the slope of that function. Let’s review how that works.

The Derivative of a Function is its Slope

Consider a function $f(x)$ . Recall the derivative is defined as

$\frac{df}{dx} = \lim\limits_{\Delta x \rightarrow \infty} \frac{f(x+\Delta x)-f(x)}{\Delta x}$ .

Let’s go through through the connection with slope slowly:

Our final sketch from the video above is critical for our understanding of derivatives and calculus in general. To that end, here’s a prettified version of the sketch that you should have in your head at all times:

Note that if $f$ is increasing as you move to the right, the slope is positive. That is,

A positive slope is upwards and to the right.

Of course, a negative slope moves down and to the right, implying the function $f$ is decreasing.

Since velocity is the derivative of position, and a derivative measures the slope of a function, it follows that

velocity is the slope of a curve showing position vs time.

Exercise 9.2: Velocity is the Derivative of Position

The three graphs below record the position of a car as a function of time. Each of the three text bubbles describe the corresponding motions in plain English.

Drag each bubble to the graph it is describing.

Let’s try a few more exercises to make sure we got it.

Exercise 9.3: Reading Position vs. Time Graphs

Drag each bubble to the graph it is describing.

Exercise 9.4:

The graph below records the position of a car as a function of time. Use the graph to answer the following questions.

Exercise 9.5:

Key Takeaways

9 Reading Velocity from Position Graphs

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