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Chapter 16: Problem 33

A particle moves in a velocity field \(\mathbf{V}(x, y)=\left\langle x^{2}, x+y^{2}\right\rangle\) If it is at position \((2,1)\) at time \(t=3,\) estimate its location at time \(t=3.01 .\)

Short Answer

Expert verified
The particle's estimated position at \(t=3.01\) is \((2.04, 1.03)\).

Step by step solution

01

Identify Given Information

We have a velocity field \( \mathbf{V}(x, y) = \left\langle x^2, x + y^2 \right\rangle \). The initial position of the particle is given as \((2, 1)\) at time \( t = 3 \). We need to estimate the position at \( t = 3.01 \).
02

Calculate Velocity at Given Position

Substitute \( x = 2 \) and \( y = 1 \) into the velocity field to find the velocity at \( t = 3 \):\[\mathbf{V}(2, 1) = \left\langle 2^2, 2 + 1^2 \right\rangle = \left\langle 4, 3 \right\rangle. \]
03

Determine Change in Position Over Small Time Interval

The small time interval is \( \Delta t = 3.01 - 3 = 0.01 \). Use the velocity to estimate the change in position: \[ \Delta \mathbf{r} = \left\langle 4 \times 0.01, 3 \times 0.01 \right\rangle = \left\langle 0.04, 0.03 \right\rangle. \]
04

Estimate New Position of the Particle

The original position of the particle at \( t = 3 \) is \( (2, 1) \). Add the change in position to the original position to find the estimated new position at \( t = 3.01 \):\[ (2, 1) + (0.04, 0.03) = (2.04, 1.03). \]
05

Conclude the Estimation

Based on the velocity and the small time change, the estimated new position of the particle at \( t = 3.01 \) is \( (2.04, 1.03) \).

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Key Concepts

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

Position Estimation
Position estimation is crucial when predicting where an object will be at a future time, especially in a dynamic environment like a velocity field. Let's think about a particle moving through space. We know its current position, but we want to predict where it will be shortly. We use a combination of mathematical models and current state information to make this estimate.
In the problem, the initial position is \(2, 1\) at \(t = 3\). This is our starting point for calculating where the particle will be at \(t = 3.01\).
  • Start by understanding the current state: You know the initial position and the velocity field.
  • Use these to calculate any change in position over a very short time, which helps improve accuracy.
Position estimation helps us to make short-term predictions using known data and mathematical transformations, crucial for planning and navigation.
Velocity Calculation
Velocity calculation is key in determining how fast something is moving in a particular direction. In our exercise, we are given a velocity field \(\mathbf{V}(x, y) = \left\langle x^2, x + y^2 \right\rangle\) that provides us the velocity vector at any point \(x, y\). This vector indicates the direction and speed.
  • To find the particle鈥檚 velocity at a specific position, substitute the position coordinates into the velocity field.
  • In our exercise, for position \(2, 1\), substituting gives \(\left\langle 4, 3 \right\rangle\).
  • This tells us that at position (2, 1), the particle moves with velocity 4 in the x-direction and 3 in the y-direction.
Accurately calculating velocity at each point enables us to precisely predict future positions by defining how far and fast something travels in each direction.
Change in Position
To estimate how much a particle's position changes over a short time, we use the velocity at its current position. \(\Delta \textbf{r}\), the change in position, is calculated by multiplying the velocity vector by the time interval \(\Delta t\).
For example, the particle's position changes as follows:
  • Given velocity vector \(\left\langle 4, 3 \right\rangle\) and \(\Delta t = 0.01\).
  • Calculate change in x: \(4 \times 0.01 = 0.04\).
  • Calculate change in y: \(3 \times 0.01 = 0.03\).
Finally, add this change to the initial position \(\left(2, 1\right)\) to find the new position \(\left(2.04, 1.03\right)\). This calculation, though over a short interval, accurately shows how far and in what direction the particle has traveled from its original position, thus offering a reliable prediction of its location.

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Most popular questions from this chapter

Find the work done by the force field $$ \mathbf{F}(x, y)=x \mathbf{i}+(y+2) \mathbf{j} $$ in moving an object along an arch of the cycloid $$\mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j} \quad 0 \leqslant t \leqslant 2 \pi$$

Find the area of the surface. The surface \(z=\frac{2}{3}\left(x^{3 / 2}+y^{3 / 2}\right), 0 \leqslant x \leqslant 1,0 \leqslant y \leqslant 1\).

Determine whether the points \(P\) and \(Q\) lie on the given surface. $$ \begin{array}{l}{\mathbf{r}(u, v)=\langle u+v, u-2 v, 3+u-v\rangle} \\ { P(4,-5,1), Q(0,4,6)}\end{array} $$

Find (a) the curl and (b) the divergence of the vector field. $$ \mathbf{F}(x, y, z)=\ln (2 y+3 z) \mathbf{i}+\ln (x+3 z) \mathbf{j}+\ln (x+2 y) \mathbf{k} $$

Let \(f\) be a scalar field and \(\mathbf{F}\) a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar field or a vector field. $$ \begin{array}{ll}{\text { (a) curl } f} & {\text { (b) grad } f} \\ {\text { (c) div } \mathbf{F}} & { \text { (d) curl(grad }f)} \\ {\text { (e) grad } \mathbf{F}} & { \text { (f) grad(div }\mathbf{F})} \\ { \text { (g) div(grad }f)} & { \text { (h) grad(div }f)} \\ { \text { (i) curl(curl }\mathbf{F})} & { \text { (j) div(div }\mathbf{F})} \\ {\text { (k) }(\operatorname{grad} f) \times(\operatorname{div} \mathbf{F})} & {\text { (1) } \operatorname{div}(\text { curl }(\text { grad } f))}\end{array} $$
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