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Chapter 11: Problem 36

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 \(t \geq 0\). $$\mathbf{a}(t)=\left\langle e^{-t}, 1\right\rangle,\left\langle u_{0}, v_{0}\right\rangle=\langle 1,0\rangle,\left\langle x_{0}, y_{0}\right\rangle=\langle 0,0\rangle$$

Short Answer

Expert verified
Question: Find the velocity and position vectors for \(t \geq 0\) given the acceleration vector \(\mathbf{a}(t) = \left\langle e^{-t}, 1 \right\rangle\), the initial velocity vector \(\mathbf{v}(0) = \left\langle 1, 0 \right\rangle\), and the initial position vector \(\mathbf{r}(0) = \left\langle 0, 0 \right\rangle\). Answer: The velocity and position vectors for \(t \geq 0\) are given by: $$ \mathbf{v}(t) = \left\langle -e^{-t}+1, t \right\rangle $$ $$ \mathbf{r}(t) = \left\langle e^{-t}+t-1, \dfrac{t^2}{2} \right\rangle $$

Step by step solution

01

Integrate the acceleration vector

To find the velocity vector, we need to integrate the acceleration vector component-wise, i.e., we integrate each component of the acceleration vector with respect to time. $$ \mathbf{v}(t) = \int \mathbf{a}(t) dt = \int \left\langle e^{-t}, 1 \right\rangle dt $$
02

Add the initial velocity vector

Now, we add the initial velocity vector \(\begin{pmatrix}u_{0}\\v_{0}\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}\) to the result of step 1. $$ \mathbf{v}(t) = \int \left\langle e^{-t}, 1 \right\rangle dt + \left\langle 1, 0 \right\rangle = \left\langle -e^{-t}+C_1, t+C_2 \right\rangle $$ Using the initial condition \(\mathbf{v}(0)=\begin{pmatrix}1\\0\end{pmatrix}\), we can find the constants \(C_1 = 1\) and \(C_2 = 0\). So, the velocity vector is: $$ \mathbf{v}(t) = \left\langle -e^{-t}+1, t \right\rangle $$
03

Integrate the velocity vector

To find the position vector, we need to integrate the velocity vector component-wise, i.e., we integrate each component of the velocity vector with respect to time. $$ \mathbf{r}(t) = \int \mathbf{v}(t) dt = \int \left\langle -e^{-t}+1, t \right\rangle dt $$
04

Add the initial position vector

Now, we add the initial position vector \(\begin{pmatrix}x_{0}\\y_{0}\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}\) to the result from step 3. $$ \mathbf{r}(t) = \int \left\langle -e^{-t}+1, t \right\rangle dt + \left\langle 0, 0 \right\rangle = \left\langle e^{-t}+t+C_3, \dfrac{t^2}{2} + C_4 \right\rangle $$ Using the initial condition \(\mathbf{r}(0)=\begin{pmatrix}0\\0\end{pmatrix}\), we can find the constants \(C_3 = -1\) and \(C_4 = 0\). So, the position vector is: $$ \mathbf{r}(t) = \left\langle e^{-t}+t-1, \dfrac{t^2}{2} \right\rangle $$ Finally, the velocity and position vectors for \(t\geq0\) are given by: $$ \mathbf{v}(t) = \left\langle -e^{-t}+1, t \right\rangle $$ $$ \mathbf{r}(t) = \left\langle e^{-t}+t-1, \dfrac{t^2}{2} \right\rangle $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Acceleration Vector
Imagine you're in a car that's speeding up; that change in speed you feel is acceleration. In vector calculus, the acceleration vector quantifies not just how fast an object is speeding up or slowing down, but also the direction it's moving in.

In our exercise, the acceleration vector is given by \( \mathbf{a}(t) = \left\langle e^{-t}, 1 \right\rangle \). This means that the acceleration in the x-direction decreases exponentially as time goes on, while in the y-direction it's steady. These components can describe, for example, a vehicle decelerating in one direction (like braking while driving east), while maintaining a constant speed in the other direction (like moving north at a steady pace).
Velocity Vector
If acceleration tells us how our speed is changing, the velocity vector describes how fast and in what direction we're moving right now. To find it, we essentially 'sum up' the acceleration over time, a process known as integration.

In the given problem, we integrated the acceleration vector \( \mathbf{a}(t) \) over time to find the velocity vector \( \mathbf{v}(t) = \left\langle -e^{-t}+1, t \right\rangle \). This tells us that the vehicle's velocity in the x-direction starts fast and slows down, while in the y-direction it steadily increases.
Position Vector
Now, where exactly is our car? The position vector answers this by describing the location of an object in space relative to a starting point, which we call the origin. To get this vector, we again use integration, this time summing up the velocity vector. Through integration, we look at the total 'distance' traveled in terms of velocity, over time.

In this exercise, after integrating the velocity vector, we obtained the position vector \( \mathbf{r}(t) = \left\langle e^{-t}+t-1, \dfrac{t^2}{2} \right\rangle \), which tells us the car's precise location at any time.
Integration of Vectors
You might be wondering, how do we get from acceleration to velocity, and from velocity to position? The answer lies in integration of vectors. This mathematical process is like adding up small pieces to get the whole. When we integrate the acceleration vector over time, like we did step by step in our solution, we're accumulating small changes in speed to find out how fast we're going. Then, integrating the velocity tells us how far we've traveled from our starting point, giving us the position vector.
Initial Conditions
Our journey needs a starting point, and that's exactly what initial conditions give us. They define where we begin in terms of velocity and position at time zero (the moment we start observing our object). In our case, the initial velocity was \( \left\langle 1, 0 \right\rangle \) and the initial position was \( \left\langle 0, 0 \right\rangle \).

With initial conditions, we can make sure that the results of our integrations actually match reality. They allow us to find the constants that arise when we integrate, such as \( C_1 \) and \( C_2 \) for velocity, and \( C_3 \) and \( C_4 \) for position, making sure our car is exactly where we expect it to be when we start the timer.

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Most popular questions from this chapter

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. Distributive properties a. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+2 \mathbf{u} \cdot \mathbf{v}+|\mathbf{v}|^{2}\) b. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+|\mathbf{v}|^{2}\) if \(\mathbf{u}\) is orthogonal to \(\mathbf{v}\) c. Show that \((\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v})=|\mathbf{u}|^{2}-|\mathbf{v}|^{2}\)

A pair of nonzero vectors in the plane is linearly dependent if one vector is a scalar multiple of the other. Otherwise, the pair is linearly independent. a. Which pairs of the following vectors are linearly dependent and which are linearly independent: \(\mathbf{u}=\langle 2,-3\rangle\) \(\mathbf{v}=\langle-12,18\rangle,\) and \(\mathbf{w}=\langle 4,6\rangle ?\) b. Geometrically, what does it mean for a pair of nonzero vectors in the plane to be linearly dependent? Linearly independent? c. Prove that if a pair of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is linearly independent, then given any vector \(w\), there are constants \(c_{1}\) and \(c_{2}\) such that \(\mathbf{w}=c_{1} \mathbf{u}+c_{2} \mathbf{v}\)

A 100-kg object rests on an inclined plane at an angle of \(30^{\circ}\) to the floor. Find the components of the force perpendicular to and parallel to the plane. (The vertical component of the force exerted by an object of mass \(m\) is its weight, which is \(m g\), where \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity.)

Hexagonal circle packing The German mathematician Gauss proved that the densest way to pack circles with the same radius in the plane is to place the centers of the circles on a hexagonal grid (see figure). Some molecular structures use this packing or its three-dimensional analog. Assume all circles have a radius of 1 and let \(\mathbf{r}_{i j}\) be the vector that extends from the center of circle \(i\) to the center of circle \(j,\) for \(i, j=0,1, \ldots, 6\) a. Find \(\mathbf{r}_{0 j},\) for \(j=1,2, \ldots, 6\) b. Find \(\mathbf{r}_{12}, \mathbf{r}_{34},\) and \(\mathbf{r}_{61}\) c. Imagine circle 7 is added to the arrangement as shown in the figure. Find \(\mathbf{r}_{07}, \mathbf{r}_{17}, \mathbf{r}_{47},\) and \(\mathbf{r}_{75}\)

An ant walks due east at a constant speed of \(2 \mathrm{mi} / \mathrm{hr}\) on a sheet of paper that rests on a table. Suddenly, the sheet of paper starts moving southeast at \(\sqrt{2} \mathrm{mi} / \mathrm{hr} .\) Describe the motion of the ant relative to the table.
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