Question

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance \(s\) from the origin is the given function. $$ s=2 \sin 3 t \cos 3 t $$

Short answer

Amplitude: 1, Period: \frac{\pi}{3}, Frequency: \frac{3}{\pi}, Velocity Amplitude: 6.

Step-by-step solution

Simplify the given function

Use the trigonometric identity \(\text{2sinAcosB = sin(A+B) + sin(A-B)}\) to simplify the given function. Here, \(A = 3t\) and \(B = 3t\). Therefore: \[s=2 \sin 3t \cos 3t = \sin(6t).\]

Identify the amplitude

The simplified function \[s = \sin(6t)\] is a standard sinusoidal function, \y = A\sin(Bt)\, where \(A = 1\) is the amplitude. Thus, the amplitude is 1.

Determine the period

The period of a sinusoidal function \y = A\sin(Bt)\ is given by \(\frac{2\pi}{B}\).\ Here, \(B = 6\), so the period is: \[\text{Period} = \frac{2\pi}{6} = \frac{\pi}{3}.\]

Calculate the frequency

The frequency is the reciprocal of the period. Therefore, \(\text{Frequency} = \frac{1}{\text{Period}}\).\ So, \[ \text{Frequency} = \frac{1}{\left(\frac{\pi}{3}\right)} = \frac{3}{\pi}.\]

Find the velocity function

The velocity is the derivative of the position function \(s\). Differentiate \[s = \sin(6t):\] \[v = \frac{ds}{dt} = 6 \cos(6t).\]

Determine the velocity amplitude

The amplitude of the velocity function \[v = 6 \cos(6t)\] is 6. Hence, the velocity amplitude is 6.

Key concepts

Amplitude

In the context of harmonic motion, the amplitude refers to the maximum extent of displacement from the equilibrium position. For a sinusoidal wave in the form \(y = A \sin(Bt) \), \(A\) represents the amplitude. This specifies the peak value that the wave can reach. In our example, the function was simplified to \(s = \sin(6t)\). Here, the amplitude \(A\) is 1 because \(\sin(6t)\) oscillates between -1 and 1. Thus, the amplitude of the motion is 1.

To visualize, if you plotted this function, you'd see waves oscillating equally above and below the horizontal axis, but never going beyond 1 or below -1. Amplitude is a crucial notion because it tells us how far the particle moves away from its central position.

Period

The period of a sinusoidal function represents the time it takes to complete one full cycle of motion. For any function of the form \(y = A \sin(Bt) \), the period \(T\) is calculated using the formula \(T = \frac{2\pi}{B}\). In our simplified function \(s = \sin(6t)\), the value of \(B\) is 6. Thus, the period is \(T = \frac{2\pi}{6} = \frac{\pi}{3}\).

What this means is that every \(\frac{\pi}{3}\) units of time, the particle returns to its initial position and state of motion. Understanding the period helps in predicting the motion pattern of the particle over time. It tells us that the particle completes one oscillation in \(\frac{\pi}{3}\) units of time.

Frequency

Frequency refers to how many complete cycles or oscillations occur in a unit of time. It is the reciprocal of the period. For the function \(s = \sin(6t)\), we determined that the period is \(\frac{\pi}{3}\). To find the frequency \(f\), we use the formula \(f = \frac{1}{T}\). Therefore, in our example, \(f = \frac{3}{\pi}\).

This tells us that the particle completes \(\frac{3}{\pi}\) oscillations per unit of time. Higher frequency indicates more rapid oscillations, while lower frequency means fewer oscillations in the same span of time. Understanding frequency helps us comprehend how often the particle in motion crosses a certain point.

Velocity Amplitude

The velocity amplitude is the maximum speed that the particle reaches during its motion. To find this, differentiate the position function with respect to time. For the function \(s = \sin(6t)\), the velocity \(v\) is \(v = \frac{ds}{dt} = 6 \cos(6t)\). Here, the velocity amplitude is the coefficient in front of the cosine function, which is 6.

So the velocity amplitude is 6. This indicates that the maximum speed the particle reaches during its motion is 6 units per time interval. It is essential for understanding how fast the particle can move and how quickly it changes direction.