Positive Acceleration Intervals For Y = 7sin(π/4 * X) In The First 10 Seconds

In the realm of physics, understanding the motion of objects is paramount. Acceleration, the rate of change of velocity, plays a crucial role in describing this motion. This article delves into the analysis of an object's acceleration, specifically focusing on identifying the intervals during which the acceleration is positive. We will explore a scenario where the acceleration y, measured in meters per second squared (m/s²), of an object after x seconds is given by the equation:

y = 7sin(π/4 * x)

Our goal is to determine the time intervals within the first 10 seconds when the object experiences positive acceleration.

Understanding the Acceleration Equation

The equation y = 7sin(π/4 * x) represents a sinusoidal function, which is characterized by its oscillating nature. Let's break down the components of this equation to gain a better understanding:

• y: Represents the acceleration of the object in meters per second squared (m/s²).
• x: Represents the time elapsed in seconds.
• 7: Represents the amplitude of the sine wave, which indicates the maximum acceleration the object experiences.
• sin(π/4 * x): Represents the sine function, which oscillates between -1 and 1. The argument (π/4 * x) affects the period and frequency of the oscillation.
• π/4: This coefficient within the sine function determines the period of the oscillation. The period is the time it takes for the sine wave to complete one full cycle. In this case, the period is calculated as 2π / (π/4) = 8 seconds. This means the acceleration pattern repeats every 8 seconds.

The sine function, sin(θ), is positive in the first and second quadrants of the unit circle (0 < θ < π). Therefore, the acceleration y will be positive when the argument of the sine function, (π/4) * x, falls within these quadrants. Understanding sinusoidal motion is crucial for various applications, from analyzing oscillations in mechanical systems to understanding wave phenomena in physics.

Determining Intervals of Positive Acceleration

To find the intervals where the acceleration is positive, we need to solve the inequality:

0 < (π/4) * x < π

Dividing all parts of the inequality by π/4, we get:

0 < x < 4

This tells us that the acceleration is positive during the time interval 0 < x < 4 seconds. However, since the sine function is periodic, the acceleration will also be positive in subsequent intervals. To find the next interval within the first 10 seconds, we can add the period (8 seconds) to the initial interval:

0 + 8 < x < 4 + 8

8 < x < 12

Since we are only considering the first 10 seconds, the interval 8 < x < 12 is partially within our range of interest. The portion of this interval that falls within the first 10 seconds is 8 < x < 10. Thus, the acceleration is positive during the time interval 8 < x < 10 seconds within the first 10 seconds.

Therefore, within the first 10 seconds, the acceleration is positive over the intervals 0 < x < 4 seconds and 8 < x < 10 seconds. These positive acceleration intervals are crucial for understanding the object's motion. During these times, the object is speeding up in the direction of its velocity.

Graphical Representation

To further visualize this, we can sketch a graph of the acceleration function y = 7sin(π/4 * x) over the interval 0 ≤ x ≤ 10. The graph will show a sinusoidal wave with a period of 8 seconds and an amplitude of 7. The portions of the graph that lie above the x-axis represent the intervals where the acceleration is positive. This graphical analysis provides a clear visual representation of the acceleration's behavior over time.

[Imagine a graph here, showing a sine wave with a period of 8 seconds and amplitude 7. The areas above the x-axis between 0 and 4 seconds, and between 8 and 10 seconds, would be shaded to indicate positive acceleration.]

Implications and Applications

Understanding the intervals of positive acceleration is crucial in various applications. For instance, in vehicle dynamics, knowing when a car is accelerating positively helps engineers design control systems that enhance performance and safety. In robotics, controlling the acceleration of robotic arms is essential for precise movements and tasks. The principles discussed here extend beyond simple object motion, finding relevance in fields such as signal processing and electrical engineering, where sinusoidal functions model oscillating signals and currents.

The concepts of acceleration and motion are fundamental to classical mechanics. Analyzing the motion of objects requires a clear understanding of how acceleration, velocity, and displacement are related. The sinusoidal nature of the acceleration in this scenario provides a valuable case study for understanding oscillatory motion, a ubiquitous phenomenon in physics and engineering.

Further Exploration

This analysis can be extended to explore other aspects of the object's motion. For example, we could determine the intervals where the acceleration is negative, indicating deceleration or slowing down. We could also calculate the object's velocity and displacement by integrating the acceleration function. Furthermore, we could investigate the effects of varying the amplitude and period of the acceleration function on the object's motion. Such explorations provide a more comprehensive understanding of kinematics and dynamics, the branches of physics concerned with motion and forces.

Conclusion

In conclusion, we have determined that the object's acceleration is positive during the intervals 0 < x < 4 seconds and 8 < x < 10 seconds within the first 10 seconds. This analysis involved understanding the properties of sinusoidal functions and solving inequalities. By identifying these intervals, we gain valuable insights into the object's motion and its behavior over time. This understanding has broad implications in various fields, from engineering to physics, where the principles of acceleration and motion are fundamental. This analysis of positive acceleration intervals serves as a cornerstone for predicting and controlling the motion of objects in diverse applications.