The Equation of a Sinusoidal Function

Oct 13, 2025 by miodeya.com 110 views

Introduction

In the world of physics and engineering, understanding the optimal angle of launch is crucial for achieving maximum range and accuracy. Kari and Samantha's water-balloon launcher is a great example of this concept. By determining the optimal angle of launch, they can ensure that their launcher performs at its best. In this article, we will explore the equation that can be used to determine the minimum and maximum optimal angles of launch.

The Equation of a Sinusoidal Function

The equation that can be used to determine the minimum and maximum optimal angles of launch is a sinusoidal function. A sinusoidal function is a mathematical function that describes a periodic oscillation. In this case, the sinusoidal function represents the relationship between the angle of launch and the range of the launcher.

The general form of a sinusoidal function is:

y = A sin(Bx) + C

where A is the amplitude, B is the frequency, and C is the vertical shift.

However, in the case of the water-balloon launcher, the equation can be simplified to:

y = sin(x)

where y is the range of the launcher and x is the angle of launch.

Finding the Minimum and Maximum Optimal Angles

To find the minimum and maximum optimal angles of launch, we need to find the values of x that correspond to the minimum and maximum values of y.

Since the equation is a sinusoidal function, the minimum and maximum values of y occur when the sine function is equal to -1 and 1, respectively.

Therefore, we can set up the following equations:

-1 = sin(x) 1 = sin(x)

To solve for x, we can use the inverse sine function:

x = arcsin(-1) x = arcsin(1)

Using a calculator, we can find the values of x:

x ≈ -90° x ≈ 90°

However, since the angle of launch cannot be negative, we can ignore the negative value.

Therefore, the minimum and maximum optimal angles of launch are:

x ≈ 90° x ≈ 45°

The Relationship Between the Angle of Launch and the Range of the Launcher

The equation y = sin(x) shows that the range of the launcher is directly proportional to the sine of the angle of launch. This means that as the angle of launch increases, the range of the launcher also increases.

However, the range of the launcher is not directly proportional to the angle of launch. Instead, it is proportional to the sine of the angle of launch.

This is because the sine function is a periodic function that oscillates between -1 and 1. As the angle of launch increases, the sine function oscillates between -1 and 1, resulting in a range of values for the launcher.

The Importance of Understanding the Optimal Angle of Launch

Understanding the optimal angle of launch is crucial for achieving maximum range and accuracy. By determining the minimum and maximum optimal angles of launch, Kari and Samantha can ensure that their water-balloon launcher performs at its best.

In addition, understanding the optimal angle of launch can also help in designing and building more efficient and effective launchers. By optimizing the angle of launch, designers and engineers can create launchers that are more accurate and have a longer range.

Conclusion

In conclusion, the equation y = sin(x) can be used to determine the minimum and maximum optimal angles of launch for a water-balloon launcher. By understanding the relationship between the angle of launch and the range of the launcher, designers and engineers can create more efficient and effective launchers.

References

[1] "Trigonometry" by Michael Corral
[2] "Physics for Scientists and Engineers" by Paul A. Tipler and Gene Mosca

Glossary

Amplitude: The maximum value of a sinusoidal function.
Frequency: The number of oscillations of a sinusoidal function per unit of time.
Vertical shift: The value that is added to the amplitude of a sinusoidal function to shift it vertically.
Inverse sine function: A mathematical function that returns the angle whose sine is a given value.
Periodic function: A mathematical function that oscillates between a set of values.