91Ó°ÊÓ

\(\mathbf{r}(t)\) is the position of a particle in the \(x y\) -plane at time \(t .\) Find an equation in \(x\) and \(y\) whose graph is the path of the particle. Then find the particle's velocity and acceleration vectors at the given value of \(t\). $$\mathbf{r}(t)=e^{t} \mathbf{i}+\frac{2}{9} e^{2} \mathbf{j}, \quad t=\ln 3$$

A formula for the curvature of a parametrized plane curve a. Show that the curvature of a smooth curve \(\mathbf{r}(t)=f(t) \mathbf{i}+\) \(g(t)\) j defined by twice-differentiable functions \(x=f(t)\) and \(y=g(t)\) is given by the formula $$ \kappa=\frac{|\dot{x} \ddot{y}-\ddot{y} \dot{x}|}{\left(\dot{x}^{2}+\dot{y}^{2}\right)^{3 / 2}} $$ The dots in the formula denote differentiation with respect to \(t\) one derivative for each dot. Apply the formula to find the curvatures of the following curves. b. \(\mathbf{r}(t)=t \mathbf{i}+(\ln \sin t) \mathbf{j}, \quad 0 < t < \pi\) \(\mathbf{c .} \mathbf{r}(t)=\left[\tan ^{-1}(\sinh t)\right] \mathbf{i}+(\ln \cosh t) \mathbf{j}\)

In Exercises \(11-14,\) find the arc length parameter along the curve from the point where \(t=0\) by evaluating the integral $$s=\int_{0}^{t}|\mathbf{v}(\tau)| d \tau$$ from Equation ( 3 ). Then find the length of the indicated portion of the curve. $$\mathbf{r}(t)=(4 \cos t) \mathbf{i}+(4 \sin t) \mathbf{j}+3 t \mathbf{k}, \quad 0 \leq t \leq \pi / 2$$

Show that a moving particle will move in a straight line if the normal component of its acceleration is zero.

A particle traveling in a straight line is located at the point (1,-1,2) and has speed 2 at time \(t=0 .\) The particle moves toward the point (3,0,3) with constant acceleration \(2 \mathbf{i}+\mathbf{j}+\mathbf{k}\) Find its position vector \(\mathbf{r}(t)\) at time \(t\)

If a string wound around a fixed circle is unwound while held taut in the plane of the circle, its end \(P\) traces an involute of the circle. In the accompanying figure, the circle in question is the circle \(x^{2}+y^{2}=1\) and the tracing point starts at ( 1,0 ). The unwound portion of the string is tangent to the circle at \(Q,\) and \(t\) is the radian measure of the angle from the positive \(x\) -axis to segment \(O Q .\) Derive the parametric equations $$x=\cos t+t \sin t, \quad y=\sin t-t \cos t, \quad t>0$$ of the point \(P(x, y)\) for the involute. (GRAPH CAN'T COPY).

Rounding the answers to four decimal places, use a CAS to find \(\mathbf{v}, \mathbf{a}\) speed, \(\mathbf{T}, \mathbf{N}, \mathbf{B}, \kappa,\) and the tangential and normal components of acceleration for the curves at the given values of \(t\) $$\mathbf{r}(t)=\left(3 t-t^{2}\right) \mathbf{i}+\left(3 t^{2}\right) \mathbf{j}+\left(3 t+t^{3}\right) \mathbf{k}, \quad t=1$$

Find an equation for the circle of curvature of the curve \(\mathbf{r}(t)=\) \((2 \ln t) \mathbf{i}-[t+(1 / t)] \mathbf{j}, e^{-2} \leq t \leq e^{2},\) at the point (0,-2) where \(t=1\)

Limits of cross products of vector functions Suppose that \(\mathbf{r}_{1}(t)=f_{1}(t) \mathbf{i}+f_{2}(t) \mathbf{j}+f_{3}(t) \mathbf{k}, \mathbf{r}_{2}(t)=g_{1}(t) \mathbf{i}+g_{2}(t) \mathbf{j}+g_{3}(t) \mathbf{k}\) \(\lim _{t \rightarrow t_{0}} \mathbf{r}_{1}(t)=\mathbf{A},\) and \(\lim _{t \rightarrow t_{0}} \mathbf{r}_{2}(t)=\mathbf{B} .\) Use the determinant formula for cross products and the Limit Product Rule for scalar functions to show that $$\lim _{t \rightarrow t_{0}}\left(\mathbf{r}_{1}(t) \times \mathbf{r}_{2}(t)\right)=\mathbf{A} \times \mathbf{B}$$