91Ó°ÊÓ

Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ Find the acceleration vector and show that its direction is always toward the center of the circle.

(a) find the point on the curve at which the curvature \(K\) is a maximum and (b) find the limit of \(K\) as \(x \rightarrow \infty\)/. $$ y=(x-1)^{2}+3 $$

In Exercises \(53-56,\) find \(\mathbf{r}(t)\) for the given conditions. $$ \mathbf{r}^{\prime}(t)=4 e^{2 t} \mathbf{i}+3 e^{t} \mathbf{j}, \quad \mathbf{r}(0)=2 \mathbf{i} $$

Find the tangential and normal components of acceleration for a projectile fired at an angle \(\theta\) with the horizontal at an initial speed of \(v_{0}\). What are the components when the projectile is at its maximum height?

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The definite integral of a vector-valued function is a real number.

Prove that if \(\mathbf{r}\) is a vector-valued function that is continuous at \(c,\) then \(\|\mathbf{r}\|\) is continuous at \(c\).

Frictional Force \(\quad\) A 5500 -pound vehicle is driven at a speed of 30 miles per hour on a circular interchange of radius 100 feet. To keep the vehicle from skidding off course, what frictional force must the road surface exert on the tires?

You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function \(\mathrm{r}\). Let \(r=\|\mathbf{r}\|,\) let \(G\) represent the universal gravitational constant, let \(M\) represent the mass of the sun, and let \(m\) represent the mass of the planet. Prove that \(\mathbf{r} \cdot \mathbf{r}^{\prime}=r \frac{d r}{d t}\)

You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function \(\mathrm{r}\). Let \(r=\|\mathbf{r}\|,\) let \(G\) represent the universal gravitational constant, let \(M\) represent the mass of the sun, and let \(m\) represent the mass of the planet. Prove Kepler's First Law: Each planet moves in an elliptical orbit with the sun as a focus.