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Chapter 10: Problem 16
Find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=\langle\arcsin t, \arccos t, 0\rangle $$
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Use the model for projectile motion, assuming there is no air resistance. A baseball, hit 3 feet above the ground, leaves the bat at an angle of \(45^{\circ}\) and is caught by an outfielder 3 feet above the ground and 300 feet from home plate. What is the initial speed of the ball, and how high does it rise?
The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=3 t \mathbf{i}+t \mathbf{j}+\frac{1}{4} t^{2} \mathbf{k} $$
The \(z\) -component of the derivative of the vector-valued function \(\mathbf{u}\) is 0 for \(t\) in the domain of the function. What does this information imply about the graph of \(\mathbf{u}\) ?
Consider the motion of a point (or particle) on the circumference of a rolling circle. As the circle rolls, it generates the cycloid \(\mathbf{r}(t)=b(\omega t-\sin \omega t) \mathbf{i}+b(1-\cos \omega t) \mathbf{j}\) where \(\omega\) is the constant angular velocity of the circle and \(b\) is the radius of the circle. Find the velocity and acceleration vectors of the particle. Use the results to determine the times at which the speed of the particle will be (a) zero and (b) maximized.
Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} \\ \mathbf{v}(1)=5 \mathbf{j}, \quad \mathbf{r}(1)=\mathbf{0} \end{array} $$
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