Question

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

Short answer

Answer: For \(t \geq 0\), the velocity vector is \(\mathbf{v}(t) = \left\langle -\cos t + 1, \sin t + 2, t\right\rangle\) and the position vector is \(\mathbf{r}(t) = \left\langle -\sin t + t, -\cos t + 2t + 1, \frac{1}{2}t^2\right\rangle\).

Step-by-step solution

Integrate the acceleration vector to find the velocity vector

In order to find the velocity vector, we will integrate the acceleration vector componentwise. The acceleration vector is given as \(\mathbf{a}(t) = \langle\sin t, \cos t, 1\rangle\): $$\begin{array}{l} \mathbf{v}(t) = \int \mathbf{a}(t) dt = \left\langle \int \sin t dt, \int \cos t dt, \int 1 dt\right\rangle \end{array}$$ Now, integrate each component: $$\begin{array}{l} \mathbf{v}(t) = \left\langle -\cos t + C_1, \sin t + C_2, t + C_3\right\rangle \end{array}$$ We are given the initial velocity vector \(\left\langle u_{0}, v_{0}, w_{0}\right\rangle = \langle 0, 2, 0\rangle\). Using this information, we can plug in \(t=0\) to our velocity vector to find the constants \(C_1\), \(C_2\), and \(C_3\): $$\begin{array}{l} \left\langle 0, 2, 0\right\rangle = \left\langle -\cos(0) + C_1, \sin(0) + C_2, 0 + C_3\right\rangle \\\ \left\langle 0, 2, 0\right\rangle = \left\langle -1 + C_1, 0 + C_2, 0 + C_3\right\rangle \end{array}$$ From here, we can easily solve for the constants: \(C_1 = 1\), \(C_2 = 2\), and \(C_3 = 0\). So, our velocity vector is: $$\begin{array}{l} \mathbf{v}(t) = \left\langle -\cos t + 1, \sin t + 2, t\right\rangle \end{array}$$

Integrate the velocity vector to find the position vector

Now that we have the velocity vector, we will integrate it componentwise to find the position vector: $$\begin{array}{l} \mathbf{r}(t) = \int \mathbf{v}(t) dt = \left\langle \int (-\cos t + 1) dt, \int (\sin t + 2) dt, \int t dt\right\rangle \end{array}$$ Integrate each component: $$\begin{array}{l} \mathbf{r}(t) = \left\langle -\sin t + t + C_4, -\cos t + 2t + C_5, \frac{1}{2}t^2 + C_6\right\rangle \end{array}$$ We are given the initial position vector \(\left\langle x_{0}, y_{0}, z_{0}\right\rangle = \langle 0, 0, 0\rangle\). Using this information, we can plug in \(t=0\) to our position vector to find the constants \(C_4\), \(C_5\), and \(C_6\): $$\begin{array}{l} \left\langle 0, 0, 0\right\rangle = \left\langle -\sin(0) + 0 + C_4, -\cos(0) + 0 + C_5, \frac{1}{2}(0)^2 + C_6\right\rangle \\\ \left\langle 0, 0, 0\right\rangle = \left\langle 0 + C_4, -1 + C_5, 0 + C_6\right\rangle \end{array}$$ From here, we can easily solve for the constants: \(C_4 = 0\), \(C_5 = 1\), and \(C_6 = 0\). So, our position vector is: $$\begin{array}{l} \mathbf{r}(t) = \left\langle -\sin t + t, -\cos t + 2t + 1, \frac{1}{2}t^2\right\rangle \end{array}$$ In conclusion, for \(t \geq 0\), the velocity vector is \(\mathbf{v}(t) = \left\langle -\cos t + 1, \sin t + 2, t\right\rangle\) and the position vector is \(\mathbf{r}(t) = \left\langle -\sin t + t, -\cos t + 2t + 1, \frac{1}{2}t^2\right\rangle\).

Key concepts

Acceleration Vector

In physics and mathematics, the acceleration vector represents the change in velocity of an object with respect to time. It is a vector function that describes the rate of change of the velocity vector for a moving object. If we know the acceleration vector, \( \mathbf{a}(t) \) as a function of time, it describes how fast and in what direction the velocity of the object changes.

For example, in the given exercise, the acceleration vector is \( \mathbf{a}(t) = \langle\sin t, \cos t, 1\rangle \). To move from acceleration to velocity, we integrate this vector with respect to time. By doing this, we can gain insight into the object's changing speed.

• Positive components of the acceleration vector indicate increasing velocity in that direction.
• Negative components suggest a reduction in speed or a reversal of direction.

It is essential to comprehend that the acceleration vector can itself be variable, which adds layers of complexity to motion in physics.

Velocity Vector

The velocity vector, \( \mathbf{v}(t) \), describes the speed and direction of an object's motion and is the derivative of the position vector concerning time. If acceleration is constant, the velocity vector is a simple linear function of time. However, if acceleration is a function of time as shown in our example, we must integrate the acceleration vector to devise the velocity.

Starting from the acceleration vector \( \mathbf{a}(t) \), we arrived at the velocity vector by integrating component-wise and then determining the constants from the given initial conditions. The ability to extract these details from an initial value problem like the initial velocity \( \left\langle u_{0}, v_{0}, w_{0}\right\rangle \) ensures that the integrated function corresponds to the physical scenario we are analyzing.

Integrating Vectors

The process of integrating vectors is central in moving from acceleration to velocity, and then from velocity to position in vector calculus. Each vector component is integrated separately, following the rules of integral calculus. This is typically done when dealing with motion in multiple dimensions.

By integrating component-wise, we have found the general form of the velocity and the position vectors. The next step is to apply the initial conditions, which act as the 'integration constants,' to specify the particular motion of the object. This process reflects the practical aspect where we know an object's initial states and want to predict its future behavior over time.

Initial Value Problem

An initial value problem in vector calculus involves finding a function that passes through a given initial point and follows a prescribed derivative function. This method is crucial in physics for predicting the future movement of objects when forces and initial states are known.

By incorporating the initial conditions, like the initial velocity and position in our example, we solved for the integration constants, ensuring that our general solution fits the particular scenario. The initial value problem is foundational in kinetics, allowing us to deduce the entirety of an object's path from just its initial impulses and conditions.