Motion In Space

(requires JavaScript)

1. The following table gives coordinates of a particle moving through space along a smooth curve. Find the average velocities over the time intervals $\left[0,1\right]$, $\left[0.5,1\right]$, $\left[1,2\right]$, and $\left[1,1.5\right]$. Estimate the velocity and speed of the particle at $t=1$.

   $t$	$x$	$y$	$z$
   $0.0$	$2.7$	$9.8$	$3.7$
   $0.5$	$3.5$	$7.2$	$3.3$
   $1.0$	$4.5$	$6.0$	$3.0$
   $1.5$	$5.9$	$6.4$	$2.8$
   $2.0$	$7.3$	$7.8$	$2.7$

   Average velocities are $⟨1.8,-3.8,-0.7⟩$, $⟨2,-2.4,-0.6⟩$, $⟨2.8,1.8,-0.3⟩$, and $⟨2.8,0.8,-0.4⟩$ respectively. When $t=1$, velocity is $2.4\mathbf{i}-0.8\mathbf{j}-0.5\mathbf{k}$ and speed is $2.58$.

2. The figure shows the path of a particle that moves with position vector $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)$ at time $t$.

   

   1. Draw a vector that represents the average velocity of the particle over the time interval $2\le t\le 2.4$.
   2. Draw a vector that represents the average velocity of the particle over the time interval $1.5\le t\le 2$
   3. Write an expression for the velocity vector $\mathbf{v}<!--FUNCTION\; APPLICATION-->\left(2\right)$.
   4. Draw an approximation to the velocity vector $\mathbf{v}<!--FUNCTION\; APPLICATION-->\left(2\right)$ and estimate the speed of the particle at $t=2$.

3. Find the velocity, acceleration, and speed of an electron with the position function

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=e^t\mathbf{i}+e^{2t}\mathbf{j}$

   Sketch the path of the electron and draw the velocity and acceleration vectors for $t=0$.

   $⟨e^t,2e^{2t}⟩$, $⟨e^t,4e^{2t}⟩$, $e^t\sqrt{1+4e^{2t}}$

4. Find the velocity, acceleration, and speed of an electron with the position function

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=t\mathbf{i}+2\cos <!--FUNCTION\; APPLICATION-->\left(t\right)\mathbf{j}+\sin <!--FUNCTION\; APPLICATION-->\left(t\right)\mathbf{k}$

   Sketch the path of the electron and draw the velocity and acceleration vectors for $t=0$.

   $⟨1,-2\sin <!--FUNCTION\; APPLICATION-->\left(t\right),\cos <!--FUNCTION\; APPLICATION-->\left(t\right)⟩$, $⟨0,-2\cos <!--FUNCTION\; APPLICATION-->\left(t\right),-\sin <!--FUNCTION\; APPLICATION-->\left(t\right)⟩$, $\sqrt{2+3{\sin }^2<!--FUNCTION\; APPLICATION-->\left(t\right)}$

5. The position function of a neutrino is given by

   $\mathbf{r}<!--FUNCTION\; APPLICATION-->\left(t\right)=⟨t^2,5t,t^2-16t⟩$

   When is the speed at minimum?

   $\mathbf{v}<!--FUNCTION\; APPLICATION-->\left(t\right)=⟨2t,5,2t-16⟩$,

   so the speed is $\left|\mathbf{v}<!--FUNCTION\; APPLICATION-->\left(t\right)\right|=\sqrt{8t^2-64t+281}$.

   The function under the root sign is a parabola, and its minimal value is at the vertex $t=4$.

6. A ball is thrown at an angle of $45°$ to the horizontal ground. If the ball lands $90$ m away, what was the initial speed of the ball?
   $30$ m/s.