Particle Motion: Velocity And Acceleration Calculation

by Alex Johnson 55 views

Understanding the motion of particles is a fundamental concept in physics and mathematics. This article will guide you through calculating velocity and acceleration from a given position function. We'll break down the steps, making it easy to follow along, even if you're new to calculus. Let's dive in and explore how to determine a particle's velocity and acceleration given its position function.

Understanding Position, Velocity, and Acceleration

Before we jump into the calculations, let's clarify the core concepts. The position of a particle, denoted as s(t), describes its location at a specific time t. Think of it as a snapshot of where the particle is on a number line or in space at that moment. Position is the foundation upon which we build our understanding of motion.

Velocity, on the other hand, tells us how the particle's position is changing over time. It's the rate of change of position and is found by taking the derivative of the position function. The velocity, represented as v(t), not only gives us the speed of the particle but also the direction in which it's moving. It's a crucial concept in understanding the dynamics of motion.

Finally, acceleration describes how the velocity of the particle is changing over time. It's the rate of change of velocity and is calculated by taking the derivative of the velocity function (or the second derivative of the position function). Acceleration, denoted as a(t), helps us understand if the particle is speeding up, slowing down, or changing direction. It adds another layer of depth to our understanding of motion.

In essence, these three concepts are interconnected. Position is the starting point, velocity is the rate of change of position, and acceleration is the rate of change of velocity. By understanding these relationships, we can analyze and predict the motion of particles in various scenarios.

(a) Finding the Velocity Function v(t)

The first part of our problem is to find the velocity function, v(t), given the position function s(t) = 7tĀ³ + 4t + 9. Remember, velocity is the rate of change of position with respect to time. This means we need to find the derivative of the position function. Differentiation is a fundamental tool in calculus that allows us to determine the instantaneous rate of change of a function. By applying differentiation to the position function, we can unlock the secrets of the particle's velocity.

To find the derivative, we'll use the power rule, which states that if f(t) = atāæ, then f'(t) = natāæā»Ā¹. This rule is the cornerstone of differentiating polynomial functions like the one we have here. It provides a systematic way to find the rate of change of each term in the position function.

Let's apply the power rule to each term in s(t):

  • The derivative of 7tĀ³ is 3 * 7tĀ² = 21tĀ².
  • The derivative of 4t is 1 * 4tā° = 4 (since tā° = 1).
  • The derivative of the constant term 9 is 0 (because constants don't change with time).

Now, we combine these results to get the velocity function:

v(t) = 21tĀ² + 4

This equation tells us the velocity of the particle at any given time t. It's a powerful tool that allows us to analyze the particle's motion in detail. The velocity function captures the essence of how the particle's position is changing over time, providing a dynamic view of its movement.

So, the velocity at time t is v(t) = 21tĀ² + 4 m/s. We've successfully found the velocity function by applying the principles of calculus and the power rule of differentiation. This is a significant step in understanding the particle's motion.

(b) Calculating the Velocity at t = 3 seconds

Now that we have the velocity function, v(t) = 21tĀ² + 4, the next step is to calculate the velocity at a specific time, t = 3 seconds. This will give us a concrete understanding of how fast the particle is moving at that particular moment. Evaluating the velocity function at a specific time provides a snapshot of the particle's motion, allowing us to pinpoint its speed and direction at that instant.

To do this, we simply substitute t = 3 into our velocity function:

v(3) = 21(3)Ā² + 4

First, we calculate 3Ā², which is 9.

v(3) = 21 * 9 + 4

Next, we multiply 21 by 9, which gives us 189.

v(3) = 189 + 4

Finally, we add 4 to 189, resulting in 193.

v(3) = 193 m/s

Therefore, the velocity of the particle at t = 3 seconds is 193 meters per second. This tells us that at 3 seconds, the particle is moving quite rapidly. The magnitude of the velocity gives us the speed, and the sign (which is positive in this case) indicates the direction of motion.

This calculation demonstrates the power of the velocity function. By simply plugging in a specific time, we can instantly determine the particle's velocity at that moment. This is a crucial skill in analyzing and predicting the motion of objects in various contexts.

(c) Determining the Acceleration Function a(t)

The final part of our problem is to find the acceleration function, a(t). Remember, acceleration is the rate of change of velocity with respect to time. This means we need to find the derivative of the velocity function, v(t) = 21tĀ² + 4. Just as we used differentiation to find velocity from position, we now use it again to find acceleration from velocity.

We'll once again use the power rule for differentiation. Applying the power rule to each term in v(t):

  • The derivative of 21tĀ² is 2 * 21tĀ¹ = 42t.
  • The derivative of the constant term 4 is 0.

Combining these results, we get the acceleration function:

a(t) = 42t

This equation tells us the acceleration of the particle at any given time t. It's a linear function, which means the acceleration is changing at a constant rate. The acceleration function provides insights into how the particle's velocity is changing over time, whether it's speeding up, slowing down, or maintaining a constant speed.

So, the acceleration at time t is a(t) = 42t m/sĀ². This means that the particle's acceleration is directly proportional to time. As time increases, the acceleration also increases. This is a key piece of information in understanding the overall motion of the particle.

By finding the acceleration function, we've completed our analysis of the particle's motion. We now have a comprehensive understanding of its position, velocity, and acceleration at any given time. This knowledge allows us to make predictions about the particle's future motion and to analyze its behavior in various scenarios.

Conclusion

In this article, we've explored how to calculate velocity and acceleration from a given position function. We started by understanding the fundamental concepts of position, velocity, and acceleration, and how they relate to each other. We then applied the power rule of differentiation to find the velocity function from the position function and the acceleration function from the velocity function. Finally, we calculated the velocity at a specific time, t = 3 seconds, to gain a concrete understanding of the particle's motion at that instant.

Understanding these concepts is crucial in many fields, including physics, engineering, and mathematics. The ability to analyze motion and predict the behavior of objects is a valuable skill that can be applied to a wide range of problems.

For further exploration of calculus and its applications in physics, consider visiting Khan Academy's Calculus section.