Preview Activity 10.3.1.
In Preview Activity 10.1.1 and Preview Activity 10.2.1 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.
(a)
In your testing center (an abandoned parking garage), you drive your car up a ramp 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 is the displacement of your total trip.
(b)
Do you think the distance the car actually traveled is greater than, less than, or exactly the same as your answer to part a? Write a couple of sentences to explain your reasoning.
(c)
In order to get more information for your telemetry system, you decide to pull the location data from your drive up the parking ramp every 10 seconds. The LTS gives the following data:
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\(\langle 10,-5,0\rangle\) at \(t=0\)
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\(\langle 6,0,6 \rangle\) at \(t=10\)
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\(\langle -10,2,10\rangle\) at \(t=20\)
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\(\langle -8,10,12\rangle\) at \(t=30\)
Calculate the distance between successive times and estimate how far your car went on the drive up the parking ramp.
(d)
Do you think your estimate from part c is greater than, less than, or equal to the actual distance traveled? Write a couple of sentences to explain your reasoning.
(e)
You look into the documentation on your location tracking system and see that you can specify how often the software will output location data. You decide that getting the location every half second seems like a good idea. How many data points will that correspond to for your drive up the ramp? Write a couple of sentences to describe what steps you would take with this new location data to calculate a better estimate on the distance traveled by your car. How might you go about finding the exact distance traveled by your car?
