Problem 13 - Calculus III

Question

A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector \(\mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k}\). The path is similar to that of a helix, although it is not a helix. The graph is shown here:

Step-by-Step Solution

Answer

The path is a spiral in the xy-plane with quadratic ascent in z, resembling an upward accelerating helix.

Step 1: Understand the Problem

We are given a position vector \( \mathbf{r}(t)=(3 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+t^{2} \mathbf{k} \) which describes the path of a person on a hang glider. Our goal is to analyze this vector equation and understand the path it describes in three dimensions.

Step 2: Analyze Components of the Vector

The position vector \( \mathbf{r}(t) = (3 \cos t) \mathbf{i} + (3 \sin t) \mathbf{j} + t^{2} \mathbf{k} \) can be broken down into three components: \( x(t) = 3 \cos t \), \( y(t) = 3 \sin t \), and \( z(t) = t^{2} \). The \( x \) and \( y \) components describe a circle of radius 3, creating a circular motion on the xy-plane, while the \( z \) component shows quadratic upward motion.

Step 3: Identify the Path Characteristics

The components \( x(t) = 3 \cos t \) and \( y(t) = 3 \sin t \) describe a circle, making the path of motion in the xy-plane circular. The \( z(t) = t^2 \) component indicates the person is rising as \( t \) increases at a rate determined by \( t^2 \), which means the person is spiraling upwards at an increasing speed, but not at a constant rate in the z-direction.

Step 4: Compare with Helix

A standard helix would have a linear component for \( z \), typically \( z(t) = kt \) for some constant \( k \), indicating a constant rate of elevation change. However, since our equation has \( z(t) = t^2 \), the rate at which height is increasing is variable, it accelerates upwards, differentiating from a standard helix.

Step 5: Conclusion

The path described by the vector \( \mathbf{r}(t) \) is a spiral path that circles in the xy-plane with a steadily increasing radius, while the height increases quadratically over time. This path resembles a helix with an upward accelerating motion due to the \( t^2 \) term in z-direction.

Key Concepts

Position Vector

The concept of a position vector is crucial in understanding motion within a coordinate system. In three-dimensional space, a position vector identifies a point or the end of a path made by a moving object relative to the origin of the space. For the journey described in this exercise, the position vector is given by \( \mathbf{r}(t) = (3 \cos t) \mathbf{i} + (3 \sin t) \mathbf{j} + t^{2} \mathbf{k} \). Here, the components of the vector \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) represent the unit directions corresponding to the x, y, and z axes respectively.
The essence of this vector is that it plays a pivotal role in pinpointing the hang glider's exact location as time progresses. Each of its components - \( 3 \cos t \), \( 3 \sin t \), and \( t^2 \) - specifies how the x, y, and z coordinates change over time. This paints a complete picture of the path in three dimensions.

Circular Motion

When examining the path of the hang glider in the xy-plane, the components \( x(t) = 3 \cos t \) and \( y(t) = 3 \sin t \) illustrate circular motion. This type of motion occurs when an object moves along a path in the shape of a circle, consistently maintaining a specific radius from a central point.
In our scenario:

• The center of the circle is at the origin (0, 0),
• The radius of the circle is 3, as both the \( x \) and \( y \) expressions are multiplied by 3,
• The path traced creates a consistent loop as \( t \) changes, maintaining a rotational motion.

While this explains motion only on the xy-plane, it demonstrates how the hang glider continuously follows a looping path within that plane.

Quadratic Upward Motion

The upward motion of the hang glider isn't just straightforward; it involves a precise quadratic nature as time \( t \) progresses. The z-component, represented by \( z(t) = t^2 \), introduces quadratic upward motion, which essentially suggests a type of acceleration.

This concept can be further understood by analyzing why this motion is called 'quadratic':

• As time increases, the square of \( t \) grows larger at a faster rate,
• This results in a rise in height that accelerates over time, not moving uniformly as it would with a linear function,
• Thus, the hang glider ascends ever faster in the z-direction, differentiating the path from a standard spiral where z increases at a fixed rate.

This non-linear climb distinguishes the path from a helix, making the motion unique and dynamic as the hang glider continually gains altitude.