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Chapter 12: Problem 35

Given an acceleration vector, initial velocity $\left\langle u_{0}, v_{0}\right\rangle,\( and initial position \)\left\langle x_{0}, y_{0}\right\rangle,\( find the velocity and position vectors, for \)t \geq 0$. $$\mathbf{a}(t)=\langle\cos t, 2 \sin t\rangle,\left\langle u_{0}, v_{0}\right\rangle=\langle 0,1\rangle,\left\langle x_{0}, y_{0}\right\rangle=\langle 1,0\rangle$$

Short Answer

Expert verified
The velocity vector is given by \(\mathbf{v}(t) = \left\langle \sin t, -2\cos t +1 \right\rangle\) and the position vector is given by \(\mathbf{r}(t) = \left\langle -\cos t + 1, -2\sin t + t \right\rangle\) for \(t \geq 0\).

Step by step solution

01

To find the velocity vector, we need to integrate the given acceleration vector with respect to time. We have $$\mathbf{a}(t) = \left\langle \cos t, 2\sin t\right\rangle$$ To find the velocity vector \(\mathbf{v}(t)\), we need to integrate components of the acceleration vector separately: $$\mathbf{v}(t) = \left\langle \int \cos t \, dt, \int 2\sin t \, dt\right\rangle$$ #Step 2: Determine the constants of integration for the velocity vector#

We found the expression for the velocity vector without the constants of integration. Now, we need to use the initial velocity vector to determine the constants of integration. The initial velocity vector is 麓\(\langle 0, 1 \rangle\). When \(t=0\): $$\mathbf{v}(0) = \left\langle \int_0^0 \cos t \, dt + C_1, \int_0^0 2\sin t \, dt + C_2\right\rangle = \langle 0,1\rangle$$ From this, we can determine \(C_1\) and \(C_2\), which are both 0. #Step 3: Express the complete velocity vector as a function of time#
02

Considering that both constants of integration are 0, our velocity vector becomes: $$\mathbf{v}(t) = \left\langle \int \cos t \, dt, \int 2\sin t \, dt\right\rangle = \left\langle \sin t, -2\cos t + 1\right\rangle$$ #Step 4: Integrate the velocity vector to find the position vector#

Now that we have the velocity vector, we need to integrate it to find the position vector \(\mathbf{r}(t)\). We can integrate components of the velocity vector separately: $$\mathbf{r}(t) = \left\langle \int \sin t \, dt, \int (-2\cos t + 1) \, dt\right\rangle$$ #Step 5: Determine the constants of integration for the position vector#
03

Similarly, as we did for the velocity vector, we need to determine the constants of integration for the position vector. The initial position vector is \(\langle 1, 0 \rangle\). When \(t=0\): $$\mathbf{r}(0) = \left\langle \int_0^0 \sin t \, dt + D_1, \int_0^0 (-2\cos t + 1) \, dt + D_2\right\rangle = \langle 1, 0\rangle$$ From this, we can determine \(D_1 = 1\) and \(D_2 = 0\). #Step 6: Express the complete position vector as a function of time#

Considering the values for the constants of integration, our position vector becomes: $$\mathbf{r}(t) = \left\langle \int \sin t \, dt + 1, \int (-2\cos t + 1)\,dt\right\rangle = \left\langle -\cos t+1, -2\sin t+t\right\rangle$$ Therefore, the velocity vector is given by \(\mathbf{v}(t) = \left\langle \sin t, -2\cos t +1 \right\rangle\) and the position vector is given by \(\mathbf{r}(t) = \left\langle -\cos t + 1, -2\sin t + t \right\rangle\) for \(t \geq 0\).

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Most popular questions from this chapter

Evaluate the following limits. $$\lim _{t \rightarrow \pi / 2}\left(\cos 2 t \mathbf{i}-4 \sin t \mathbf{j}+\frac{2 t}{\pi} \mathbf{k}\right)$$

Determine whether the following statements are true and give an explanation or counterexample. a. The line \(\mathbf{r}(t)=\langle 3,-1,4\rangle+t\langle 6,-2,8\rangle\) passes through the origin. b. Any two nonparallel lines in \(\mathbb{R}^{3}\) intersect. c. The curve \(\mathbf{r}(t)=\left\langle e^{-t}, \sin t,-\cos t\right\rangle\) approaches a circle as \(t \rightarrow \infty\). d. If \(\mathbf{r}(t)=e^{-t^{2}}\langle 1,1,1\rangle\) then \(\lim _{t \rightarrow \infty} \mathbf{r}(t)=\lim _{t \rightarrow-\infty} \mathbf{r}(t)\).

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3,0\rangle, \mathbf{v}=\langle 1,1,1\rangle\)

A pair of lines in \(\mathbb{R}^{3}\) are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect. determine the point(s) of intersection. $$\begin{array}{l} \mathbf{r}(t)=\langle 4+t,-2 t, 1+3 t\rangle ;\\\ \mathbf{R}(s)=\langle 1-7 s, 6+14 s, 4-21 s\rangle \end{array}$$

A baseball leaves the hand of a pitcher 6 vertical feet above home plate and \(60 \mathrm{ft}\) from home plate. Assume that the coordinate axes are oriented as shown in the figure. a. In the absence of all forces except gravity, assume that a pitch is thrown with an initial velocity of \(\langle 130,0,-3\rangle \mathrm{ft} / \mathrm{s}\) (about \(90 \mathrm{mi} / \mathrm{hr}\) ). How far above the ground is the ball when it crosses home plate and how long does it take for the pitch to arrive? b. What vertical velocity component should the pitcher use so that the pitch crosses home plate exactly \(3 \mathrm{ft}\) above the ground? c. A simple model to describe the curve of a baseball assumes that the spin of the ball produces a constant sideways acceleration (in the \(y\) -direction) of \(c \mathrm{ft} / \mathrm{s}^{2}\). Assume a pitcher throws a curve ball with \(c=8 \mathrm{ft} / \mathrm{s}^{2}\) (one-fourth the acceleration of gravity). How far does the ball move in the \(y\) -direction by the time it reaches home plate, assuming an initial velocity of (130,0,-3) ft/s? d. In part (c), does the ball curve more in the first half of its trip to the plate or in the second half? How does this fact affect the batter? e. Suppose the pitcher releases the ball from an initial position of (0,-3,6) with initial velocity \((130,0,-3) .\) What value of the spin parameter \(c\) is needed to put the ball over home plate passing through the point (60,0,3)\(?\)
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