Problem 27
Question
The velocity of a particle moving in a fluid is described by means of a velocity field \(\mathbf{v}=v_{1} \mathbf{i}+v_{2} \mathbf{j}+v_{3} \mathbf{k}\), where the components \(v_{1}, v_{2}\), and \(v_{3}\) are functions of \(x, y, z\), and time \(t\). If the velocity of the particle is \(\mathbf{v}(t)=6 t^{2} x \mathbf{i}-4 t y^{2} \mathbf{j}+2 t(z+1) \mathbf{k}\) find \(\mathbf{r}(t) .\)
Step-by-Step Solution
Verified Answer
Position function: \( \mathbf{r}(t) = (x_0 + 2t^3)\mathbf{i} + (y_0 - 2ty^2)\mathbf{j} + (z_0 + tz + t)\mathbf{k} \).
1Step 1: Understanding the Problem
We need to find the position vector \( \mathbf{r}(t) \) of a particle given the velocity field \( \mathbf{v}(t)=6t^{2}x \mathbf{i} - 4ty^{2} \mathbf{j} + 2t(z+1) \mathbf{k} \). The position vector is typically obtained by integrating the velocity function with respect to time.
2Step 2: Integrate the x-Component of Velocity
We have the velocity component \( v_1 = 6t^2 x \). However, to integrate it, we need the form \( \frac{dx}{dt} = v_1(t) \). This confirms the x-component depends only on time and can be integrated as \( x(t) = x_0 + \int 6t^2 dt = x_0 + 2t^3 + C_1 \), where \( x_0 \) and \( C_1 \) are constants.
3Step 3: Integrate the y-Component of Velocity
For the component \( v_2 = -4ty^2 \), treat it similarly: \( \frac{dy}{dt} = -4ty^2 \). Integrate with respect to time: \( y(t) = y_0 - 2ty^2 + C_2 \), where \( y_0 \) and \( C_2 \) are integration constants.
4Step 4: Integrate the z-Component of Velocity
The component \( v_3 = 2t(z+1) \) is treated as \( \frac{dz}{dt} = 2t(z+1) \). Integrate to find \( z(t) = z_0 + t(z+1) + C_3 \), again accounting for the constant \( z_0 \) and \( C_3 \).
5Step 5: Combine Results to Get Position Vector
Now combine these integrated forms to construct the position vector \( \mathbf{r}(t) = (x(t), y(t), z(t)) \). Using the equations from the previous steps, we have \[ \mathbf{r}(t) = (x_0 + 2t^3 + C_1) \mathbf{i} + (y_0 - 2ty^2 + C_2) \mathbf{j} + (z_0 + t(z+1) + C_3) \mathbf{k} \].
Key Concepts
Position VectorIntegrationVelocity ComponentsParticle Motion
Position Vector
In mathematics and physics, the position vector is a crucial concept for describing the location of a point in space. It is typically denoted as \(\mathbf{r}(t)\), where \(t\) represents time.
The position vector provides a way to express the motion of a particle by detailing its x, y, and z coordinates as functions of time.
By understanding a particle's position vector, we can gain insight into how the particle moves in a fluid or other environment.
This vector results from integrating the particle's velocity components, providing a complete description of its path over time. In this exercise, the components derive from the velocity field given as \(\mathbf{v}(t)= 6t^{2} x \mathbf{i} - 4ty^{2} \mathbf{j} + 2t(z+1) \mathbf{k}\).
Thus, to determine \(\mathbf{r}(t)\), you integrate each velocity component with respect to time to establish each coordinate of the position vector.
The position vector provides a way to express the motion of a particle by detailing its x, y, and z coordinates as functions of time.
By understanding a particle's position vector, we can gain insight into how the particle moves in a fluid or other environment.
This vector results from integrating the particle's velocity components, providing a complete description of its path over time. In this exercise, the components derive from the velocity field given as \(\mathbf{v}(t)= 6t^{2} x \mathbf{i} - 4ty^{2} \mathbf{j} + 2t(z+1) \mathbf{k}\).
Thus, to determine \(\mathbf{r}(t)\), you integrate each velocity component with respect to time to establish each coordinate of the position vector.
Integration
Integration is a fundamental process in calculus used to find the original function from its derivative. In the context of motion and velocity, integration helps us determine the position vector from given velocity components.
The integration process involves reversing differentiation, effectively undoing it to obtain the position function from the velocity function.
In this problem, each component of the velocity field, specifically \(v_1\), \(v_2\), and \(v_3\), requires integration to ascertain the displacement in their respective directions.
The integration process involves reversing differentiation, effectively undoing it to obtain the position function from the velocity function.
In this problem, each component of the velocity field, specifically \(v_1\), \(v_2\), and \(v_3\), requires integration to ascertain the displacement in their respective directions.
- For the \(x\)-component, \(v_1 = 6t^2 x\)
- For the \(y\)-component, \(v_2 = -4ty^2\)
- For the \(z\)-component, \(v_3 = 2t(z+1)\)
Velocity Components
Each component in a velocity field represents the rate of change of position in one spatial dimension.
In this exercise, the velocity field \(\mathbf{v}(t)\) is expressed as a vector with three components: \(v_1\), \(v_2\), and \(v_3\), each a function of position and time.
These components describe how fast and in what direction the particle moves in the \(x\), \(y\), and \(z\) directions:
In this exercise, the velocity field \(\mathbf{v}(t)\) is expressed as a vector with three components: \(v_1\), \(v_2\), and \(v_3\), each a function of position and time.
These components describe how fast and in what direction the particle moves in the \(x\), \(y\), and \(z\) directions:
- \(v_1 = 6t^2 x\) is the rate of change in the x-direction.
- \(v_2 = -4ty^2\) indicates speed and direction along the y-axis.
- \(v_3 = 2t(z+1)\) describes motion in the z-direction.
Particle Motion
Particle motion is aptly described through its position vector that changes over time according to its velocity. In physics, understanding particle motion involves studying how particles travel through space.
The given velocity field captures how the particle's speed and direction vary within a fluid medium.
As time progresses, the particle experiences different accelerations and directional changes, which are encapsulated in the velocity vector \(\mathbf{v}(t)\).
By integrating the velocity components for \(x\), \(y\), and \(z\), we compute the trajectory traced by the particle over time, manifesting as the position vector \(\mathbf{r}(t)\).
The given velocity field captures how the particle's speed and direction vary within a fluid medium.
As time progresses, the particle experiences different accelerations and directional changes, which are encapsulated in the velocity vector \(\mathbf{v}(t)\).
By integrating the velocity components for \(x\), \(y\), and \(z\), we compute the trajectory traced by the particle over time, manifesting as the position vector \(\mathbf{r}(t)\).
- This integration defines its continuous motion, tracing a path determined by the given velocity expressions.
- The constants of integration \(x_0\), \(y_0\), and \(z_0\) represent initial conditions.
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