Question

A beach ball is riding the waves near Tofino, British Columbia. The ball goes up and down with the waves according to the formula \(h=1.4 \sin \left(\frac{\pi t}{3}\right),\) where \(h\) is the height, in metres, above sea level, and \(t\) is the time, in seconds. a) In the first \(10 \mathrm{s},\) when is the ball at sea level? b) When does the ball reach its greatest height above sea level? Give the first time this occurs and then write an expression for every time the maximum occurs. c) According to the formula, what is the most the ball goes below sea level?

Short answer

a) 0, 3, 6, 9 seconds. b) First at 1.5 seconds; every 6n + 1.5 seconds. c) 1.4 meters below sea level.

Step-by-step solution

Understanding the given function

The height of the ball above sea level is given by the function \(h=1.4 \sin \left(\frac{\pi t}{3}\right)\). The height, \(h\), is in meters, and time, \(t\), is in seconds.

Finding when the ball is at sea level

Sea level is defined as \(h = 0\). Set the given equation to zero: \[1.4 \sin\left(\frac{\pi t}{3}\right) = 0\]. Solve for \(t\): \[\sin\left(\frac{\pi t}{3}\right) = 0\]. The sine function equals zero at multiples of \(\pi\): \[\frac{\pi t}{3} = n\pi\], where \(n\) is an integer. Solving for \(t\) gives \[t = 3n\]. Therefore, \(t\) will be 0, 3, 6, or 9 seconds in the first 10 seconds.

Finding the greatest height

The maximum value of \(\sin(x)\) is 1. Therefore, the greatest height is when: \[1.4 \sin\left(\frac{\pi t}{3}\right) = 1.4\]. Solve for \(t\) when \(\sin\left(\frac{\pi t}{3}\right) = 1\): \[\frac{\pi t}{3} = \frac{\pi}{2} + 2n\pi\]. Solving for \(t\)\, \[t = 1.5 + 6n\], where \(n\) is an integer. The first time this occurs is when \(t = 1.5\). The expression for every time the maximum occurs is \(t = 1.5 + 6n\).

Finding the most the ball goes below sea level

The most the ball goes below sea level corresponds to the minimum value of the sine function. The minimum value of \(\sin(x)\) is -1. Therefore, the lowest height (most below sea level) is: \[1.4 \sin\left(\frac{\pi t}{3}\right) = -1.4\].

Key concepts

sinusoidal functions

In mathematics, a sinusoidal function is any function that describes a smooth, periodic oscillation. These functions are generally represented by sine or cosine curves. In our exercise, the beach ball's height is governed by the formula: \[h=1.4 \sin\left(\frac{\pi t}{3}\right)\]. The function incorporates the sine curve to determine the ball’s vertical movement over time.

Sinusoidal functions are critical in modeling real-world phenomena that cycle or repeat over time, such as waves, sound waves, and even day-night cycles. Understanding these functions involves knowing their key characteristics, such as amplitude, period, and phase shift.

* **Amplitude**: This is the height from the midpoint of the wave to its peak. For the given function, the amplitude is 1.4 meters, which means the ball moves 1.4 meters above and below the sea level.
* **Period**: This is the length of time it takes for the wave to complete one full cycle. Here, the period can be calculated using the formula for a sine function’s period: \[T=\frac{2 \pi}{\frac{\pi}{3}} = 6 \text{ seconds}\].
* **Phase Shift**: This value indicates any horizontal shift from the origin. Our function does not have a phase shift, meaning it starts its cycle at \(t = 0\). Understanding these components helps in predicting and analyzing the wave motion in various contexts.

wave height calculations

Wave height calculations are essential to determine key moments when the ball reaches specific altitudes, like sea level and its highest and lowest points.

Firstly, to find out when the ball is at sea level, we set the height \(h\) to zero: \[1.4 \sin\left(\frac{\pi t}{3}\right) = 0\]. Solving for \(t\), we find the ball is at sea level at multiples of 3 seconds, less than 10 seconds (0, 3, 6, and 9 seconds).

To calculate the time when the ball is at its greatest height, we solve for \(t\) when the sine function’s value is 1: \[\sin\left(\frac{\pi t}{3}\right) = 1\]. This equation resolves to \(t = 1.5 + 6n\) seconds. Here, the first time this maximum height is achieved is at 1.5 seconds, repeating every 6 seconds.

Lastly, the ball's most significant dip below sea level occurs when the sine function reaches -1. The lowest height is thus -1.4 meters, indicating the ball descends 1.4 meters below the sea level.

harmonic motion

Harmonic motion, often referred to as simple harmonic motion (SHM), describes the back-and-forth oscillatory movement of a system where the restoring force is proportional to the displacement. Our beach ball scenario depicts a real-world example of SHM influenced by wave motion.

In SHM, two main forces work: the restoring force and inertia. They work together to create a periodic motion. For the beach ball:

• The **restoring force** comes from the wave pulling the ball back toward the sea level after it moves away.
• **Inertia** drives the ball to continue past the equilibrium when the wave crests or troughs.

These combined forces lead to oscillations described by sinusoidal functions.

Key properties of SHM include:

• **Period**: The time for one complete cycle of motion, which in this example, is 6 seconds.
• **Amplitude**: The maximum displacement from the equilibrium point, here being 1.4 meters.
• **Frequency**: The number of oscillation cycles per unit of time. It is the inverse of the period.

Understanding harmonic motion's principles aids in comprehending wave movements in various physical systems, be it in seas, springs, or even pendulums. It explains why our beach ball follows the sinusoidal motion dictated by the waves.