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 $t$:

$\mathbf{r}(t) = \langle 2-t, 4 \sqrt{5}\rangle, t = 1$

$\mathbf{r}(t) = t \mathbf{i} + 2 \cos t \mathbf{j} + \sin t \mathbf{k}, t = 0$

Required force:

What force is required so that a particle of mass $m$ has the position function $\mathbf{r}(t) = t^3 \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k}$?

A force with magnitude 20N acts directly upward from the $xy$-plane on an object with mass 4 kg. The object starts at the origin with initial velocity $\mathbf{v}(0) = \mathbf{i}-\mathbf{j}$. Find its position function and its speed at time $t$.

Tangential and normal components of acceleration:

Find the tangential and normal components of the acceleration vector. $$ \mathbf{r}(t) = (1+t) \mathbf{i} + (t^2 -2t)\mathbf{j}.$$

Play ball!

A ball with mass 0.8 kg is thrown southward into the air with a speed of 30 m/s at an angle of $30^{\circ}$ to the ground. A west wind applies a steady force of 4 N to the ball in an easterly direction. Where does the ball land and with what speed?

Law of Conservation of Angular Momentum

If a particle with mass $m$ moves with position vector $\mathbf{r}(t)$, then its angular momentum is defined as $\mathbf{L}(t) = m \mathbf{r}(t) \times \mathbf{v}(t)$ and its torque as $\mathbf{\tau}(t) = m \mathbf{r}(t) \times \mathbf{a}(t)$. Show that $\mathbf{L}'(t) = \mathbf{\tau}(t)$. Deduce that if $\mathbf{\tau}(t) = 0$ for all $t$, then $\mathbf{L}(t)$ is constant. This is the law of conservation of angular momentum.