Chapter 10: Problem 49
In Exercises \(49-52,\) evaluate the definite integral. $$ \int_{0}^{1}(8 t \mathbf{i}+t \mathbf{j}-\mathbf{k}) d t $$
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The position vector \(r\) describes the path of an object moving in the \(x y\) -plane. Sketch a graph of the path and sketch the velocity and acceleration vectors at the given point. $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j},(3,0) $$
Consider the vector-valued function \(\mathbf{r}(t)=\left(e^{t} \sin t\right) \mathbf{i}+\left(e^{t} \cos t\right) \mathbf{j}\). Show that \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) are always perpendicular to each other.
Find the indefinite integral. $$ \int\left[(2 t-1) \mathbf{i}+4 t^{3} \mathbf{j}+3 \sqrt{t} \mathbf{k}\right] d t $$
Use the model for projectile motion, assuming there is no air resistance. Find the vector-valued function for the path of a projectile launched at a height of 10 feet above the ground with an initial velocity of 88 feet per second and at an angle of \(30^{\circ}\) above the horizontal. Use a graphing utility to graph the path of the projectile.
Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ $$ \text { Show that the magnitude of the acceleration vector is } b \omega^{2} \text { . } $$
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