Question

Consider the motion of an object given by the position function $$ \mathbf{r}(t)=f(t)\langle a, b, c\rangle+\left\langle x_{0}, y_{0}, z_{0}\right\rangle, \text { for } t \geq 0 $$ where \(a, b, c, x_{0}, y_{0},\) and \(z_{0}\) are constants, and \(f\) is a differentiable scalar function, for \(t \geq 0\) a. Explain why this function describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?

Short answer

Question: Explain why the position function \(\mathbf{r}(t) = f(t)\langle a, b, c\rangle + \langle x_{0}, y_{0}, z_{0}\rangle\) describes motion along a line, and find the velocity function and determine if the velocity is constant in magnitude or direction along the path. Answer: a. The motion is along a line because for every \(t\geq0\), there exists a point \((x, y, z)\) on the line such that the position function satisfies the condition \(\frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c} = f(t)\). b. The velocity function is \(\mathbf{v}(t) = \frac{df}{dt}\langle a, b, c\rangle\). The velocity is constant in direction along the path, as \(\langle a, b, c\rangle\) is constant. However, the magnitude of the velocity may or may not be constant depending on the scalar function \(f(t)\).

Step-by-step solution

a. Why the motion is along a line?

To prove that the motion described by the given position function is along a line, we need to show that the vector \(\mathbf{r}(t)\) lies on a straight line for any \(t \geq 0\). Let's consider an arbitrary point \((x, y, z)\) on the line that the motion follows. If we can represent this point in the form of the given position function, then we can say that the motion is along a line. \((x, y, z) = f(t)\langle a, b, c\rangle + \langle x_{0}, y_{0}, z_{0}\rangle\) We can divide by the components, assuming none of them is zero (if one or more are zero, the argument holds with the nonzero ones): \(\frac{x-x_{0}}{a} = \frac{y-y_{0}}{b} = \frac{z-z_{0}}{c} = f(t)\) As \(f\) is a differentiable scalar function, we can say that for every \(t \geq 0\), there exists a point \((x, y, z)\) on the line such that the given position function satisfies the condition. Hence, the motion is along a line.

b. Velocity function and its properties

To find the velocity function, we need to take the derivative of the position function \(\mathbf{r}(t)\) with respect to time \(t\): \(\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t) = \frac{d}{dt}\left[f(t)\langle a, b, c\rangle + \langle x_{0}, y_{0}, z_{0}\rangle\right]\) Here, \(\langle a, b, c\rangle\) and \(\langle x_{0}, y_{0}, z_{0}\rangle\) are constants, so their derivatives with respect to time are zero. Thus, \(\mathbf{v}(t) = \frac{df}{dt}\langle a, b, c\rangle\) Now let's determine if the velocity is constant in magnitude or direction along the path. First, we find the magnitude of the velocity: \(\lVert\mathbf{v}(t)\rVert = \lVert\frac{df}{dt}\langle a, b, c\rangle\rVert = \left|\frac{df}{dt}\right|\lVert\langle a, b, c\rangle\rVert\) Since \(\lVert\langle a, b, c\rangle\rVert\) is constant, the magnitude of the velocity \(\lVert\mathbf{v}(t)\rVert\) will be constant if and only if \(\left|\frac{df}{dt}\right|\) is constant over the entire path. Similarly, the direction of the velocity vector will be constant if and only if \(\langle a, b, c\rangle\) is constant over the entire path, which is given. However, the magnitude may or may not be constant depending on the scalar function \(f(t)\). Therefore, in general, the velocity is constant in direction, but not necessarily in magnitude along the path.

Key concepts

Position Function

The position function plays a critical role in understanding motion in calculus. It represents the location of an object at any given time. In our exercise, the position function is defined as \[\mathbf{r}(t) = f(t)\langle a, b, c\rangle + \langle x_{0}, y_{0}, z_{0}\rangle\] where the constants \(a, b, c, x_{0}, y_{0},\) and \(z_{0}\) describe a fixed direction and an initial position, and \(f(t)\) is a differentiable scalar function of time \(t\).

Imagine this as if you were tracking an object's journey along a straight path. The function \(f(t)\) tells us how far along the path it is, whereas \(\langle a, b, c\rangle\) gives us the pathway's direction. The positional constants \(x_0, y_0, z_0\) mark where the object started. The fact that this function represents a line comes from the consistent direction given by the vector \(\langle a, b, c\rangle\), with movement along this line determined by the scalar \(f(t)\).

Velocity Function

A velocity function is the derivative of the position function and it represents the rate at which the object's position is changing over time. To elucidate the velocity function from the position function provided in the problem, we calculate the derivative with respect to time:

\[\mathbf{v}(t) = \frac{d}{dt}\mathbf{r}(t) = \frac{df}{dt}\langle a, b, c\rangle\]
This simplified formula indicates that the velocity at any time \(t\) depends on the derivative of the scalar function \(f'(t)\) and the constant direction \(\langle a, b, c\rangle\). It's akin to sitting in a car and measuring how fast the scenery changes as you travel down the road. If \(f'(t)\) is consistent over time, you're maintaining a steady speed; if it changes, your speed is varying.

Differentiable Scalar Function

In our context, a differentiable scalar function is a single-valued function that can be differentiated with respect to time. Differentiability ensures that the function is smooth, without any sharp corners or breaks. Here, \(f(t)\) reflects how the distance along the motion path changes with time.

One beneficial analogy is to think of the differentiable scalar function as the gas pedal's pressure in a car. Depending on how much you press down on the pedal (the value of \(f(t)\)), the car's speed will change. The derivative, \(f'(t)\), measures this change: how the pressure of your foot on the pedal is altering over time, either pushing down for more speed or easing up to slow down.

Direction and Magnitude of Velocity

The direction and magnitude of velocity are crucial for understanding motion. The direction points to where an object is heading, while magnitude shows how fast it's traveling. As seen in the motion described by this exercise, the direction of the velocity function is constant and given by the vector \(\langle a, b, c\rangle\). The magnitude of velocity, however, is determined by \(\left|\frac{df}{dt}\right|\) and the constant magnitude of the direction vector \(\langle a, b, c\rangle\).

To visualize this, imagine you're on a straight road. The road's direction doesn't change, just like our constant direction vector. However, the speed at which you drive (magnitude) can vary, much like how the rate of change of our scalar function \(f(t)\) will affect the overall speed at which the object moves along the line.