Max Displacement: Calculus And Trig Methods

Introduction

Hey guys! Today, we're diving into an exciting problem that combines the power of calculus and trigonometry. We're going to figure out how to find the maximum displacement of a mass from its starting point. This is a classic problem that pops up in physics and engineering, and it's a fantastic way to flex our math muscles. We'll be using a bit of calculus, mainly derivatives, but don't worry, we'll keep it nice and gentle. We'll also lean on our trusty trigonometric functions – sine and cosine – to help us model the motion. So, grab your thinking caps, and let's get started!

In tackling this problem of finding the maximum displacement, we're essentially trying to determine the furthest distance a mass travels from its initial position. This isn't just an abstract math problem; it has real-world applications. Imagine a pendulum swinging, a spring bouncing, or even the movement of a piston in an engine. Understanding how to calculate the maximum displacement helps us predict the behavior of these systems, ensuring their safe and efficient operation. The key here is understanding that the velocity of the mass, given by the derivative function v(t) = x'(t) = 3sin(t) + 4cos(t), holds the crucial information we need. When the velocity is zero, the mass momentarily stops before changing direction. These points of zero velocity are potential locations where the mass reaches its maximum displacement. By carefully analyzing these points and the initial conditions, we can pinpoint the furthest distance the mass travels. This process involves setting the velocity function equal to zero and solving for t, which leads us to the times when the mass is at a standstill. Then, we'll integrate the velocity function to find the position function, x(t), and evaluate it at these critical times to determine the displacement. The initial condition, which tells us the mass's position at t = 0, is also vital because it gives us a starting point to measure the displacement from. So, let's dive into the specifics and work through the steps to solve this problem together!

Problem Setup

Okay, let's break down the problem. We have a mass that's moving, and its velocity at any time t is given by the equation:

v(t) = x'(t) = 3sin(t) + 4cos(t)

This equation tells us how fast the mass is moving and in what direction at any given moment. We also know that at time t = 0, the mass is 1 unit away from its initial position. This is our starting point, our initial condition. Our goal is to find the maximum displacement of the mass from this initial position. In simpler terms, we want to know the farthest the mass travels from where it started.

The concept of displacement is crucial here. It's not just about the total distance traveled; it's about the change in position from the starting point. Think of it like this: if you walk 5 steps forward and then 5 steps back, you've traveled a total distance of 10 steps, but your displacement is zero because you're back where you started. To find the maximum displacement, we need to identify the points in time when the mass is furthest away from its initial position. These points often occur when the mass changes direction, which happens when the velocity is zero. So, our next step is to figure out when the velocity v(t) is equal to zero. This involves setting the equation 3sin(t) + 4cos(t) = 0 and solving for t. The solutions to this equation will give us the times when the mass is momentarily at rest before changing direction, and these are the key moments we need to investigate to find the maximum displacement. By combining these critical times with our initial condition, we can paint a clear picture of the mass's movement and pinpoint its furthest excursion from the starting position.

Finding Critical Points

So, the first thing we need to do is find the critical points. These are the times when the velocity is zero, as these are potential points where the displacement could be at a maximum or minimum. To do this, we set v(t) equal to zero:

3sin(t) + 4cos(t) = 0

Now, we need to solve this trigonometric equation for t. There are a few ways to approach this, but one common method is to rearrange the equation and use the tangent function. Let's subtract 4cos(t) from both sides:

3sin(t) = -4cos(t)

Next, divide both sides by cos(t) (assuming cos(t) isn't zero – we'll check that later) and by 3:

tan(t) = sin(t) / cos(t) = -4/3

Now we have a simple equation involving the tangent function. To find the values of t that satisfy this equation, we need to think about the unit circle and the properties of the tangent function. The tangent function is negative in the second and fourth quadrants. We can find a reference angle, let's call it α, such that tan(α) = 4/3. Then, the solutions for t will be in the form t = π - α + nπ, where n is an integer. This accounts for all the angles in the second and fourth quadrants that have a tangent of -4/3. We'll need to find the specific values of t that are relevant to our problem, which will depend on the context and any given time interval.

To actually find the value of α, we can use the arctangent function: α = arctan(4/3). This gives us the reference angle in the first quadrant. Then, we can use this reference angle to find the angles in the second and fourth quadrants where the tangent is -4/3. Remember, the tangent function has a period of π, so the solutions will repeat every π radians. We'll need to consider the initial condition and the specific time frame we're interested in to determine which solutions are relevant. Once we have these critical points, we can move on to finding the position function and evaluating it at these points to determine the maximum displacement. This is where the calculus comes in, as we'll need to integrate the velocity function to get the position function. But for now, we've made good progress in identifying the potential times when the mass is at its furthest point from the starting position. The solutions for t obtained here are the cornerstone for our next steps in solving this problem.

Finding the Position Function

Alright, now that we've found the critical points, the next step is to determine the position function, x(t). Remember, we know the velocity function, v(t) = x'(t) = 3sin(t) + 4cos(t). To find the position function, we need to integrate the velocity function. Integration is the reverse process of differentiation, so it will essentially