Question

Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=2 \mathbf{i}+3 \mathbf{k} \\ \mathbf{v}(0)=4 \mathbf{j}, \quad \mathbf{r}(0)=\mathbf{0} \end{array} $$

Short answer

The velocity function is \(\mathbf{v}(t) = 2t\mathbf{i} + 4\mathbf{j} + 3t\mathbf{k}\), position function is \(\mathbf{r}(t) = t^2\mathbf{i} + 4t\mathbf{j} + \frac{3}{2}t^2\mathbf{k}\), and position at time \(t=2\) is \(\mathbf{r}(2) = 4\mathbf{i} + 8\mathbf{j} + 6\mathbf{k}\)

Step-by-step solution

Calculate the velocity function

To find the velocity function \(\mathbf{v}(t)\), we integrate the acceleration function \(\mathbf{a}(t)=2 \mathbf{i}+3 \mathbf{k}\). The integral of a vector function is found by integrating each component function. This gives us: \[\mathbf{v(t)}=\int \mathbf{a}(t) dt = \int (2 \mathbf{i}+3 \mathbf{k}) dt = 2t\mathbf{i} + 3t\mathbf{k} + C\]Where \(C\) is the integration constant vector, which corresponds to the initial velocity, i.e. \(\mathbf{v}(0) = 4 \mathbf{j}\). So, our velocity function becomes:\[\mathbf{v(t)} = 2t\mathbf{i} + 4\mathbf{j} + 3t\mathbf{k}\]

Calculate the position function

\To then find the position function \(\mathbf{r}(t)\), we integrate the velocity function. We find the integral of each component, and we get: \[\mathbf{r(t)}=\int \mathbf{v}(t) dt = \int (2t\mathbf{i} + 4\mathbf{j} + 3t\mathbf{k}) dt = t^2\mathbf{i} + 4t\mathbf{j} + \frac{3}{2}t^2\mathbf{k} + D\]The integration constant \(D\) is the initial position vector, \(\mathbf{r}(0) = \mathbf{0}\). Thus, our position function becomes: \[\mathbf{r(t)} = t^2\mathbf{i} + 4t\mathbf{j} + \frac{3}{2}t^2\mathbf{k}\]

Find position at time t=2

Now we substitute \(t=2\) into the position function \(\mathbf{r}(t)\):\[\mathbf{r}(2) = (2)^2\mathbf{i} + 4*(2)\mathbf{j} + \frac{3}{2}*(2)^2\mathbf{k} = 4\mathbf{i} + 8\mathbf{j} + 6\mathbf{k}\]

Key concepts

Velocity and Position Vectors

Understanding the relationship between velocity and position vectors is a cornerstone in vector calculus, particularly in the context of motion. A velocity vector, denoted as \( \mathbf{v}(t) \), represents the rate of change of the position vector \( \mathbf{r}(t) \) with respect to time. It essentially describes the speed of an object moving along a path and the direction of the motion at any given instant.

Velocity vectors are derivatives of position vectors. If you have a position vector that tells you an object's location at any time \(t\), differentiating this position vector with respect to time yields the object's velocity. Conversely, to find the position from velocity, you integrate the velocity vector with respect to time. This forms the basis of motion analysis in physics and engineering, connecting how an object gets from point A to point B, and how quickly it does so.

To find position at a specific time, you need only substitute that time value into the position vector function. For example, given a position vector function \( \mathbf{r}(t) \) as in the exercise above, calculating the position at \(t=2\) provides us with a snapshot of where an object would be at that particular instant.

Integration of Vector Functions

Integration of vector functions is a central operation in vector calculus. It involves calculating the antiderivatives of vector functions, which is essentially a reverse process of differentiation. When you integrate acceleration, which is a vector function, you get the velocity function; integrating velocity, in turn, gives you the position function.

For each component of the vector function, you perform the integration separately. If \( \mathbf{a}(t) = 2 \mathbf{i} + 3 \mathbf{k} \), the integration process for getting velocity, and subsequently position, involves finding the indefinite integral for each scalar component function. This results in a new vector function that represents, for example, velocity or position, depending on the original function you are integrating.

Integration constants, represented by vectors \(C\) and \(D\) in the solution steps, are crucial as they embody the initial conditions of the function being integrated. These constants are found using known values at a particular time, such as the initial velocity or position.

Initial Velocity and Position

Initial velocity and position are indispensable initial conditions in vector calculus when dealing with motion. At time \(t=0\), an object has a specific starting point and begins with a certain velocity, both of which are vectors. These are represented by \( \mathbf{v}(0) \) and \( \mathbf{r}(0) \) respectively.

The initial velocity vector \( \mathbf{v}(0) \) carries information about the object's speed and direction at the very start of observation. Whereas the initial position vector \( \mathbf{r}(0) \) pinpoints the exact location of the object at this initial moment. When integrating to find a velocity or position function, these initial vectors become the integration constants. They are vital because they allow the resulting vector functions to accurately reflect the object's motion from its true starting point.

In the context of the exercise, the given initial velocity is \( 4 \mathbf{j} \) and the initial position is the zero vector, which means the object starts from the origin. Recognizing and applying these initial values helps us to accurately integrate acceleration functions to find velocity, and velocity functions to pinpoint position over time.