Rigid Body Motion Velocity Calculation At T=3 Seconds

Hey guys! Let's dive into the fascinating world of physics and explore how to describe the motion of a rigid body. We're going to break down a specific problem involving a body moving according to a given time function, and we'll calculate its velocity at a particular instant. So, buckle up and get ready to learn!

Decoding the Time Function: S = 20T - 2T²

In this scenario, motion analysis is crucial, and we're given a time function, S = 20T - 2T², which describes the position (S) of the body in meters at any given time (T) in seconds. This equation is our key to unlocking the secrets of the body's movement. It tells us how the position changes over time. The first term, 20T, represents a constant velocity component, while the second term, -2T², indicates a deceleration or a change in velocity over time. This means the body isn't moving at a constant speed; it's either speeding up or slowing down. Understanding the components of this equation is the first step in figuring out the body's motion at any specific time.

The equation represents a parabolic relationship between position and time. This is because of the T² term, which is characteristic of uniformly accelerated motion. If we were to plot this equation on a graph, with time on the x-axis and position on the y-axis, we'd see a curve resembling a parabola. The negative sign in front of the 2T² term tells us that the parabola opens downwards, indicating that the body's velocity is decreasing over time. This also means that at some point, the body will stop and potentially change direction. Visualizing this parabolic motion can help in understanding the body's overall trajectory and how its velocity changes throughout the motion. Analyzing the time function carefully allows us to predict the body's position at any given moment, and more importantly, to determine its velocity and acceleration.

The time function given, S = 20T - 2T², is a quadratic equation, and understanding its properties is essential for solving the problem. The coefficients in this equation hold significant physical meanings. The coefficient '20' in the term 20T relates to the initial velocity of the body. If the term -2T² wasn't present, the body would move with a constant velocity of 20 meters per second. However, the presence of the -2T² term introduces a changing velocity component. The coefficient '-2' in this term relates to the acceleration of the body. Specifically, it's related to half of the acceleration. To find the actual acceleration, we need to double this coefficient, giving us an acceleration of -4 meters per second squared. The negative sign indicates that the acceleration is in the opposite direction of the initial velocity, which means the body is decelerating. This deceleration is what causes the velocity to change over time, eventually leading the body to slow down and potentially reverse its direction. Recognizing these coefficients and their physical interpretations is a crucial step in analyzing the motion described by the equation.

Calculating Velocity: The Key to Unlocking Motion

Now, to find the velocity, we need to use a little calculus magic! Remember, velocity is the rate of change of position with respect to time. In mathematical terms, this means we need to find the derivative of the position function (S) with respect to time (T). The derivative of a function tells us how the function is changing at any given point. For our equation S = 20T - 2T², the derivative, which represents the velocity (V), is found by applying the power rule of differentiation. This rule states that the derivative of Tⁿ is nTⁿ⁻¹. So, let's apply this to our equation.

Taking the derivative of S = 20T - 2T² with respect to T, we get V = 20 - 4T. This new equation, V = 20 - 4T, is the velocity function. It tells us the velocity of the body at any given time T. Notice how this equation is linear, meaning that the velocity changes at a constant rate. The constant term '20' represents the initial velocity, which is the velocity at time T = 0. The term '-4T' indicates the change in velocity over time, which is influenced by the acceleration. The coefficient '-4' represents the acceleration, confirming our earlier analysis of the original time function. This velocity function is a crucial tool for understanding the body's motion because it allows us to calculate the velocity at any specific time. By simply plugging in a value for T, we can determine how fast the body is moving and in what direction at that particular instant. The process of finding the derivative is fundamental in physics for transforming position functions into velocity functions, and further into acceleration functions.

Understanding the relationship between position, velocity, and acceleration is fundamental in physics. Position describes where an object is, velocity describes how fast and in what direction it's moving, and acceleration describes how its velocity is changing. Mathematically, velocity is the first derivative of position with respect to time, and acceleration is the first derivative of velocity with respect to time (or the second derivative of position). This hierarchical relationship allows us to move from one description of motion to another. In our problem, we started with a position function, S = 20T - 2T², and we found the velocity function by taking its derivative. If we wanted to, we could further find the acceleration function by taking the derivative of the velocity function, V = 20 - 4T. This would give us a constant acceleration of -4 m/s², which confirms our previous analysis. Recognizing these relationships and how they are mathematically represented through derivatives is crucial for solving motion problems and understanding the dynamics of moving objects.

Finding the Velocity at T = 3 Seconds

Alright, the moment we've been waiting for! We have our velocity function, V = 20 - 4T. Now we just need to plug in T = 3 seconds to find the velocity at that specific time. This is a straightforward substitution. We replace T in the equation with 3 and do the math. So, V = 20 - 4(3). Let's break this down step by step to make sure we get it right. First, we multiply 4 by 3, which gives us 12. Then, we subtract 12 from 20. This gives us our final answer: V = 20 - 12 = 8 meters per second.

The calculation V = 20 - 4(3) = 8 m/s is the culmination of our analysis. This result tells us that at the instant T = 3 seconds, the body is moving with a velocity of 8 meters per second. This velocity is positive, which indicates the body is still moving in its initial direction, although it has been decelerating since the start of its motion. If we were to look at earlier times, the velocity would be higher, and at later times, the velocity would continue to decrease, possibly even becoming negative if the body were to stop and reverse its direction. The 8 m/s is a snapshot of the body's motion at a specific moment. It's crucial to understand that this is an instantaneous velocity, meaning the velocity at a single point in time, as opposed to an average velocity over a period of time. This instantaneous velocity gives us precise information about the body's state of motion at that exact instant.

So, to recap, we used the velocity function to find the velocity at a specific time. We plugged in T = 3 seconds into the equation V = 20 - 4T and calculated the result to be 8 meters per second. This answer makes sense in the context of the problem. We know the body is decelerating, so its velocity should be decreasing over time. At T = 0, the velocity was 20 m/s, and at T = 3, it's 8 m/s, which shows a decrease. This step-by-step calculation highlights the power of using mathematical equations to describe and predict motion. By understanding the relationships between position, velocity, and time, and by applying the principles of calculus, we can accurately determine the velocity of a moving object at any given moment. This approach is fundamental in physics and engineering, allowing us to analyze and design systems involving motion.

Final Answer: The Body's Velocity at 3 Seconds

Therefore, guys, at T = 3 seconds, the body's velocity is 8 meters per second. We've successfully navigated through the problem, decoded the time function, calculated the velocity function, and pinpointed the velocity at the specific instant. Physics is awesome, isn't it? Keep exploring and stay curious!

Answer: 8 m/s