13.4 Videos Guide

13.4a

• Describing motion

  • Position: $\mathbf{r}(t)$

  • Velocity: $\mathbf{v}(t)=\mathbf{r}'(t)$

  • Speed: $|\mathbf{r}'(t)|$

  • Acceleration: $\mathbf{a}(t)=\mathbf{r}''(t)$

• Equations for the motion of a projectile in $\mathbb{R}^{2}$ with initial velocity $v_{0}$ and angle $\alpha$ with the horizontal

  • $x=(v_{0}\cos\alpha)t$, $y=h_{0}+\left(v_{0}\sin\alpha\right)t-\frac{1}{2}gt^{2}$

Exercises:

13.4b

• Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of $t$.
  $\mathbf{r}(t)=\left\langle t^{2},\frac{1}{t^{2}}\right\rangle$, $t=1$

13.4c

• Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.
  $\mathbf{a}(t)=\sin t\mathbf{i}+2\cos t\mathbf{j}+6t\ \mathbf{k}$, $\mathbf{v}(0)=-\mathbf{k}$, $\mathbf{r}(0)=\mathbf{j}-4\mathbf{k}$

13.4d

• Tangential and normal components of acceleration

  • $\mathbf{a}=a_{T}\mathbf{T}+a_{N}\mathbf{N}$, where $a_{T}=\frac{\mathbf{r}'\cdot\mathbf{r}''}{\left|\mathbf{r}'\right|}$ and $a_{N}=\frac{\left|\mathbf{r}'\times\mathbf{r}''\right|}{\left|\mathbf{r}'\right|}\$

Exercises:

13.4e

• Find the tangential and normal components of the acceleration vector.
  $$\mathbf{r}(t)=t\ \mathbf{i}+2e^{t}\ \mathbf{j}+e^{2t}\mathbf{\ k}$$

13.4f

• A projectile is fired from a tank with initial sped 400 m/s. Find two angles of elevation that can be used to hit a target 300 m away.