Splitting the Acceleration Vector

Compute the splitting of the acceleration for the curve given by $\vr(t)= \langle t, \frac{t^2}{2},\frac{t^3}{6} \rangle$ at $t=1$. You should calculate $a_{\vT}$, $a_{\vN}$, $\vT$, and $\vN$. Additionally, you should verify $\va = a_{\vT} \vT +a_{\vN} \vN$ and that $\vT$ is orthogonal to $\vN$.

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 $P_1$…

1. Compute the splitting of the acceleration for the curve given by $\vr(t)= \langle t, \frac{t^2}{2},\frac{t^3}{6} \rangle$ at $t=-1$. You should calculate $a_{\vT}$, $a_{\vN}$, $\vT$, and $\vN$, then verify $\va = a_{\vT} \vT +a_{\vN} \vN$ and that $\vT$ is orthogonal to $\vN$.

2. Compute the splitting of the acceleration for the curve given by $\vr(t)= \langle t, e^t,t e^t \rangle$ at $t=0$. You should calculate $a_{\vT}$, $a_{\vN}$, $\vT$, and $\vN$, then verify $\va = a_{\vT} \vT +a_{\vN} \vN$ and that $\vT$ is orthogonal to $\vN$.

Given a curve with a parameterization such that $\vv(11)= \langle 3,4 \rangle$ and $\va(11) = \langle -2,1 \rangle$.

1. Find $\vT(11)$

2. Find $a_{\vT}(11)$

3. Find $a_{\vT}(11) \vT(11)$

4. Find $a_{\vN}(11)$

5. Find $a_{\vN}(11) \vN(11)$

6. Find $\vN(11)$

In this exercise we want to relate the scalar and vector measurements involved in the splitting of acceleration to different modes of transportation. While everyone has different experiences with piloting different modes of transportation, we will phrase this question in terms of driving a car. Feel free to substitute another mode of transportation like riding a bike, riding a scooter, driving a b…

In this exercise we want to relate the scalar and vector measurements involved in the splitting of acceleration to different modes of transportation. While everyone has different experiences with piloting different modes of transportation, we will phrase this question in terms of riding a bike. Feel free to substitute another mode of transportation like driving a car, riding a scooter, driving a b…