Exercise 4

In this exercise, we will compute and compare elements of the splitting of acceleration for three different people (Nick, Jane, and Tony) as they drive around the same racetrack.

1. Suppose that at time $t=b$, Jane is at a location which we will denote $P_1$ in her race car. The velocity and acceleration vectors for Jane are $\langle{2,0,-2}\rangle$ and $\langle{3,2,-3}\rangle$. Calculate the following:

   1. $\vec{T}(b)$
   2. $a_{\vec{T}}(b)$
   3. $a_{\vec{N}}(b)$
   4. $\vec{N}(b)$

2. Suppose that at time $t=c$, Nick is at a location $P_1$ in his race car. Without any calculation at all, answer the following:

   1. $\vec{T}(c)$
   2. $\vec{N}(c)$

3. The velocity and acceleration vectors for Nick (at time $c$) are $\langle{1,0,-1}\rangle$ and $\langle{-1,2,1}\rangle$. Calculate the following:

   1. $a_{\vec{T}}(c)$
   2. $a_{\vec{N}}(c)$

4. Is Jane using the gas pedal or the brake pedal when she is at $P_1$? Explain how you know this based on your calculations above.

5. Is Nick using the gas pedal or the brake pedal when he is at $P_1$? Explain how you know this based on your calculations above.

6. Nick and Jane talk to their friend Tony who claims that his velocity and acceleration vectors at position $P_1$ are $\langle{4,0,6}\rangle$ and $\langle{8,9,10}\rangle$. Explain how Jane and Nick know he is lying.