91Ó°ÊÓ

A projectile is fired with initial speed \(v_{0}\) feet per second from a height of \(h\) feet at an angle of \(\theta\) above the horizontal. Assuming that the only force acting on the object is gravity, find the maximum altitude, horizontal range and speed at impact. $$v_{0}=120, h=10, \theta=\frac{\pi}{3}$$

Relate to spherical coordinates defined by \(x=\rho \cos \theta \sin \phi, y=\rho \sin \theta \sin \phi\) and \(z=\rho \cos \phi,\) where \(0 \leq \rho, 0 \leq \theta \leq 2 \pi\) and \(0 \leq \phi \leq \pi.\)

Evaluate the given indefinite or definite integral. $$\int\left\langle t e^{t^{2}}, 3 t \sin t, \frac{3 t}{t^{2}+1}\right\rangle d t$$

Sketch the curve and compute the curvature at the indicated points. \(\mathbf{r}(t)=\left\langle t, t, t^{2}-1\right\rangle, t=0, t=2\)

Use graphing technology to sketch the curve traced out by the given vector- valued function. $$\mathbf{r}(t)=\left\langle t, t, 2 t^{2}-1\right\rangle$$

A projectile is fired with initial speed \(v_{0}\) feet per second from a height of \(h\) feet at an angle of \(\theta\) above the horizontal. Assuming that the only force acting on the object is gravity, find the maximum altitude, horizontal range and speed at impact. $$v_{0}=320, h=10, \theta=\frac{\pi}{4}$$

Evaluate the given indefinite or definite integral. $$\int\left\langle e^{-3 t}, t^{2} \cos t^{3}, t \cos t\right\rangle d t$$

Sketch the curve and compute the curvature at the indicated points. \(\mathbf{r}(t)=\langle 2 t-1, t+2, t-3\rangle, t=0, t=2\)

Use graphing technology to sketch the curve traced out by the given vector- valued function. $$\mathbf{r}(t)=\left\langle t^{3}-t, t^{2}, 2 t-4\right\rangle$$

A projectile is fired with initial speed \(v_{0}\) feet per second from a height of \(h\) feet at an angle of \(\theta\) above the horizontal. Assuming that the only force acting on the object is gravity, find the maximum altitude, horizontal range and speed at impact. $$\nu_{0}=240, h=10, \theta=\frac{\pi}{3}$$