2.7 Splitting the Acceleration Vector

Splitting the Acceleration Vector

Introduction

Splitting Acceleration

Efficient Calculations for the Splitting of

All your good work as CEO and lead engineer at Steer Clear has started to pay off, literally. When you showed your work on using the location tracking system (LTS) to develop navigation and telemetry tools to an investment group (your grandparents), they were impressed and decided to give you money to further develop your self-driving car. You decided to use this infusion of cash to buy a gyroscopic accelerometer which will measure acceleration as a vector with magnitude and direction. In order to use your new instrument for information on the “driving” part of your self-driving car, you need to understand how your accelerometer readings will relate to the operations needed to drive a car.

The image below shows the path you drove on a test drive along with points $P$ and $Q$ and the direction of travel. At the points $P$ and $Q$, we have drawn three vectors. The acceleration vector $\va$ provided by your new gyroscopic accelerometer is shown in magenta. The two blue vectors are the unit tangent vector $\vT$ and unit normal vector $\vN$ at that point. Label each of the blue vectors as being $\vT$ or $\vN$.

The tangential component of acceleration is defined as $a_{\vT} = \va \cdot \vT$, while equation establishes that $a_{\vT} = \frac{d\text{(speed)}}{dt}$ is also true. Use these formulas to determine whether $a_{\vT}$ can be zero. Either explain why $a_{\vT}$ is always nonzero or describe scenarios in which $a_{\vT}=0$.

Determine whether $a_{\vT}$ can be negative. Either explain $a_{\vT}$ is never negative or describe scenarios in which $a_{\vT}\lt 0$

The normal component of acceleration is defined as $a_{\vN} = \va \cdot \vN$, while equation establishes that $a_{\vN} = (\text{speed}) \vecmag{\frac{d\vT}{dt}}$ is also true. Use these formulas to determine whether $a_{\vN}$ can be negative. Either explain why $a_{\vN}$ is negative or describe scenarios in which $a_{\vN}\lt 0$.

Determine whether $a_{\vN}$ can be zero. Either explain why $a_{\vN}$ is always nonzero or describe scenarios in which $a_{\vN}=0$.

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

Pedal Usage: We measure the pedal usage as $+$, $-$, or $0$ in the following way: -$+$ if the gas pedal is being used -$-$ if the brake pedal is being used -$0$ if no pedal is being used Since our drivers are safety minded, they do not use more than one pedal at a time. Note that this measurement is a sign rather than a numerical value. Which vector calculus quantity's sign corresponds to the pedal usage? Write a sentence to explain your answer.

Describe how the value of $a_{\vec{N}}$ would be felt by the driver in our analogy.

In terms of the race car analogy, explain why $a_{\vec{N}}$ can't be negative.

Write a few sentences about whether $a_{\vT}$ is a property of the driver or a property of the road.

Write a few sentences about whether $a_{\vN}$ is a property of the driver or a property of the road.