Question

Suppose an object moves in space with the position function \(\mathbf{r}(t)=\langle x(t), y(t), z(t)\rangle .\) Write the integral that gives the distance it travels between \(t=a\) and \(t=b\)

Short answer

Answer: The integral to compute the total distance traveled by an object between time \(t=a\) and \(t=b\) is given by: $$\text{Distance} = \int_{a}^{b} |\mathbf{v}(t)|\, dt = \int_{a}^{b} \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} \,dt$$ where \(\mathbf{v}(t)\) is the velocity vector and \(\mathbf{r}(t)\) is the position function.

Step-by-step solution

Determine the velocity vector \(\mathbf{v}(t)\)

Find the derivative of the position function \(\mathbf{r}(t)\) with respect to time \(t\). This gives us the velocity vector, \(\mathbf{v}(t)=\langle x'(t), y'(t), z'(t) \rangle\). To find the derivatives, differentiate the components of \(\mathbf{r}(t)\) with respect to \(t\): \(\mathbf{v}(t) = \left\langle \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt} \right\rangle\).

Compute the magnitude of the velocity vector \(|\mathbf{v}(t)|\)

The magnitude of the velocity vector \(|\mathbf{v}(t)|\) is given by the formula: \(|\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}\).

Set up the integral for the distance traveled

To find the total distance traveled between \(t=a\) and \(t=b\), we integrate the magnitude of the velocity vector, \(|\mathbf{v}(t)|\), with respect to time \(t\). The integral is: $$\text{Distance} = \int_{a}^{b} |\mathbf{v}(t)|\, dt = \int_{a}^{b} \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} \,dt$$

Key concepts

Position Function

The concept of a position function is crucial in understanding motion in physics and math. It's a vector function that describes the specific location of an object in space relative to a defined starting point. This function depends on time and is denoted as \(\mathbf{r}(t)\), which is a three-dimensional vector for motion in space represented by a triplet of functions \(\langle x(t), y(t), z(t)\rangle\). Each component of this vector function corresponds to the position of the object along the X, Y, and Z axes at any given time \(t\).

When analyzing motion, it’s the changes in the position function over time that provide insights into the object’s behavior. As time progresses, observing how these values change can tell us the path or trajectory the object is following through space.

Velocity Vector

Moving from position to motion, we introduce the velocity vector. It's a derivative of the position function and it represents the rate of change of the object’s position with respect to time. The notation \(\mathbf{v}(t)\) corresponds to the velocity vector and implies how fast and in what direction the object is moving. It is obtained by differentiating the position function for each spatial component - essentially giving us a new vector \(\langle x'(t), y'(t), z'(t)\rangle\).

For instance, if \(x(t)\) explains the object's position along the X axis, then \(\frac{dx}{dt}\) will represent the velocity of the object in that direction. Taken together for all three dimensions, the velocity vector provides a complete description of the object's instantaneous velocity in space.

Magnitude of Velocity

To comprehend how fast an object is actually moving regardless of its direction, we look at the magnitude of velocity. It’s a scalar quantity, meaning it has only magnitude without direction, and is determined from the velocity vector. Formally, it's denoted as \(|\mathbf{v}(t)|\) and it is calculated using the Euclidean norm, which is just the square root of the sum of squared components of the velocity vector:

\[|\mathbf{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2}\]
Think of it as how you’d measure the 'straight-line' speed of an object moving diagonally - by looking at how much ground it covers irrespective of the zigzag path it might be taking. It gives us a clear measure of the object’s speed at any instance.

Definite Integral

The definite integral is a fundamental concept in calculus, remarkable for computing areas, volumes, and - in our case - the total distance traveled by an object. Essentially, the definite integral accumulates quantities across an interval. Symbolically, it's represented as \(\int_{a}^{b} f(t) dt\), where \(a\) and \(b\) define the limits of integration over which we wish to accumulate the function \(f(t)\).

When applied to the magnitude of the velocity, the definite integral sums up the infinitesimally small distances covered over each moment of time between \(t=a\) and \(t=b\), effectively giving us the total distance traveled during that time span. So, this tool not only measures static shapes but also dynamic changes over time.

Calculus

Finally, let’s embrace the broader field - calculus. It’s the mathematical study of continuous change and includes concepts such as derivatives and integrals, which play central roles in analyzing motion. Calculus bridges the gap between an abstract mathematical description and the physical reality we observe.

It’s the derivative that unveils the velocity vector and its magnitude from the position function, revealing the speed and direction of movement. The definite integral then takes these elements to calculate the total distance traveled, a tangible piece of information directly applicable to real-world scenarios. Calculus, thus, provides us the tools to decode complex motions and predict future positions and velocities, demonstrating its indispensable value in science and engineering.