2.4 Arc Length

Arc Length

Introduction

Arc Length

Moving with Unit Speed

The Driver or The Road?

In Preview Activity 2.2 and Preview Activity 2.3 we saw how the position reported by the location tracking system (LTS) of a self-driving car corresponds to a vector-valued function that describes the location of the car at a given time. We want to relate the location at different times to properties about how the car is being driven. In order to attract investors and drive shareholder value in Steer Clear, you decide to use the location tracking system's output to build a customized navigation and telemetry tool for a self-driving car. The first element you will need to build is a way to use the LTS output to calculate how far the car has driven in a given time.

In your testing center (an abandoned parking garage), you drive your car around different levels of your testing center and note that the LTS recorded that the car had coordinates of $\langle 10,-5,0\rangle$ at $t=0$ and $\langle -8,10,12\rangle$ at $t=30$ seconds. The units on each component of the coordinate vectors are meters. What is the distance between initial ($t=0$) and final ($t=30$) positions of the car? This distance is the length of the displacement vector of your total trip.

Work through this exercise and explain your reasoning step by step.

Parameterize a circle of radius $3$ centered at the origin. Give bounds on the parameter.

Use your parameterization from the previous part in the definite integral of Theorem 2.4.1 to calculate the circumference of a circle of radius $3$.

Find the exact length of the spiral defined by $\vr(t) = \langle 3\cos(t), 3\sin(t), t \rangle$ on the interval $[0,2\pi]$.

Explain why your result for the length of the spiral is larger than the circumference of the circle of the same radius.

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

Let's start by looking at a couple of easy measurements. Is the time elapsed a property of the driver or the road? Be sure to explain your answer.

Is position (the location of the car on the racetrack) a property of the driver or the road? Be sure to explain your answer.

Now that we are warmed up, let's look at some more interesting measurements. The car's speedometer reading measures how fast (as a scalar) the car is moving. Is the car's speedometer reading a property of the driver or the road? Be sure to explain your answer.

What vector calculus quantity is the speedometer reading?

The racecar's odometer measures the distance traveled by the car. Every car's odometer is set to be zero at the start of the race. Is the car's odometer reading a property of the driver or the road? Be sure to explain your answer.

Verify that the parameterization $\vr_1(s)$ of the curve $C$ in Example 2.4.3 has unit speed.

We can adapt the arc length formula to curves in 2-space that define $y$ as a function of $x$ as the following activity shows.

Let $y = f(x)$ define a smooth curve in 2-space. Parameterize this curve and use Equation  to show that the length of the curve defined by $f$ on an interval $[a,b]$ is