Problem 53 - Calculus Early Transcendentals

Question

Suppose that the equations of motion of a paper airplane during the first 12 seconds of flight are $$x=t-2 \sin t, \quad y=2-2 \cos t \quad(0 \leq t \leq 12)$$ What are the highest and lowest points in the trajectory, and when is the airplane at those points?

Step-by-Step Solution

Answer

The highest point is at \( y = 4 \) when \( t = 3\pi \), and the lowest point is at \( y = 0 \) when \( t = 0 \).

Step 1: Identify the vertical motion equation

Given the equations of motion, the vertical motion is represented by the equation \( y = 2 - 2 \cos t \). Our task is to find the highest and lowest points in this vertical trajectory.

Step 2: Analyze the vertical position function

The vertical position \( y = 2 - 2\cos t \) depends on the cosine function, which oscillates between \(-1\) and \(1\). By analyzing how \( \cos t \) influences \( y \), we can determine the highest and lowest points.

Step 3: Determine the range of the cosine function

The \( \cos t \) function ranges from \(-1\) to \(1\). Consequently, \( y \) will have a minimum when \( \cos t = 1 \) and a maximum when \( \cos t = -1 \). This allows us to find the critical points of \( y \).

Step 4: Calculate the highest point

For the highest point, set \( \cos t = -1 \). Thus, \( y_{max} = 2 - 2(-1) = 2 + 2 = 4 \). This occurs wherever \( t = \pi(2n+1) \) for integers \( n \), specifically \( t = 3\pi, 9\pi\) within the given range \([0, 12]\). However, only \( t = 9\pi/3 = 3\pi \) fits within this range.

Step 5: Calculate the lowest point

For the lowest point, set \( \cos t = 1 \). Thus, \( y_{min} = 2 - 2(1) = 2 - 2 = 0 \). This occurs wherever \( t = 2n\pi \) for integers \( n \), specifically \( t = 0, 2\pi, 4\pi,...\) within the specified range. Checking within \([0, 12]\), we find \( t = 0 \) satisfies the equation for the range provided.

Step 6: Conclusion and time calculation

The highest point of the paper airplane's trajectory is at \( y = 4 \) when \( t = 9\pi/3 = 3\pi \). The lowest point is at \( y = 0 \) when \( t = 0 \). No other values of \( t \) satisfying these conditions lie within \( 0 \leq t \leq 12 \).

Key Concepts

Motion trajectory

In this problem, the concept of motion trajectory is crucial in understanding the path taken by a moving object, such as our paper airplane. The motion is described through parametric equations, where each coordinate's motion is expressed as a function of time.
The parametric equations given are:

• Horizontal motion: \( x = t - 2 \sin t \)
• Vertical motion: \( y = 2 - 2 \cos t \)

Here, the variable \( t \) represents time in seconds. Parametric equations allow us to describe more complex paths that cannot be captured by a single equation.
The combination of horizontal and vertical motions defines the trajectory—the 2D path in the plane—of our paper airplane over the time frame \( 0 \leq t \leq 12 \). This shows how, at any given time, we can determine the position of the airplane.

Critical points

Critical points in the context of this motion refer to the specific points in time when the paper airplane reaches its maximum or minimum height. Identifying these points involves analyzing the vertical motion equation: \( y = 2 - 2 \cos t \).
The critical aspect here is understanding how the cosine function influences \( y \).

• The function \( \cos t \) oscillates between -1 and 1.
• These extremes determine the trajectory's vertical peaks: maximum and minimum.
  At \( \cos t = -1 \), \( y \) reaches its maximum value. Conversely, at \( \cos t = 1 \), \( y \) achieves its minimum value.

Thorough analysis shows that during the initial 12 seconds, the paper airplane reaches a maximum height when \( t = 3\pi \), and a minimum height at \( t = 0 \). These critical times help define the noteworthy behavior of the motion trajectory.

Trigonometric functions

Trigonometric functions are fundamental to this problem, particularly the cosine function, which significantly impacts our vertical motion equation. The rules of trigonometric functions, particularly their periodic nature, are integral in determining motion characteristics.
The equation \( y = 2 - 2 \cos t \) is pivotal because it uses the cosine function to model vertical displacement. Here is a quick overview of how this works:

• The cosine function, \( \cos t \), has a range of [-1, 1].
  When plugged into the equation, it results in the paper airplane oscillating in height.
• Understanding specific outputs of \( \cos t \) enables us to predict and calculate precise high (maximum) and low (minimum) points on the trajectory.
• Trigonometrically, these values occur at angles that are any integer multiples of \( \pi \) for minimum height and odd multiples of \( \pi \) for maximum height within the given range.

The clever use of trigonometric functions in parametric forms allows us to express complex motion with elegance and precision, helping us predict the object's path by analyzing its mathematical model.