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Chapter 24: Problem 39

The work done on an object is equal to the force times the distance moved in the direction of the force. The velocity of an object in the direction of a force is given by $$\begin{array}{ll} v=4 t & 0 \leq t \leq 4 \\ v=16+(4-t)^{2} & 4 \leq t \leq 14 \end{array}$$ where \(v=\mathrm{m} / \mathrm{s}\). Employ the multiple-application Simpson's rule to determine the work if a constant force of \(200 \mathrm{N}\) is applied for all \(t\)

Short Answer

Expert verified
In order to calculate the work done on the object, we employed multiple-application Simpson's rule to approximate the integrals of the given two velocity functions. After summing up the results from the approximations of both integrals and multiplying the result by the constant force of 200 N, we can obtain the work done on the object. The final value of the work done will depend on the chosen number of intervals (n) and might be different if a more accurate integration method is employed.

Step by step solution

01

Define the velocity functions

There are two specific velocity functions given: 1. \(v(t) = 4t \) for \(0 \leq t \leq 4\) 2. \(v(t) = 16 + (4-t)^2\) for \(4 \leq t \leq 14\)
02

Set up Simpson's rule

We will use Simpson's rule, with a given number of intervals (n) to estimate the integral of the velocity functions over the given intervals. Simpson's rule is given by: \[ f(x) dx \approx \frac{\Delta x}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + f(x_n) \right] \] For our problem, we should apply Simpson's rule separately for both velocity functions and sum up the results.
03

Approximate the integrals of the velocity functions using Simpson's rule

Let's say we use n intervals. Then, for the first velocity function: 1. \(\Delta x = \frac{4-0}{n} = \frac{4}{n}\) 2. \(x_i = 0 + i\Delta x\) 3. \(f(x_i) = 4x_i\) So, the first integral approximation using Simpson's rule is: \[ \int_0^4 4t \, dt \approx \frac{4}{3n} \left[4(0) + 4\cdot4(\Delta x) + 2\cdot4(2\Delta x) + \cdots + 4(4) \right] \] For the second velocity function: 1. \(\Delta x = \frac{14-4}{n} = \frac{10}{n}\) 2. \(x_i = 4 + i\Delta x\) 3. \(f(x_i) = 16 + (4-x_i)^2\) The second integral approximation using Simpson's rule is: \[ \int_4^{14} \left[16 + (4-t)^2\right] dt \approx \frac{10}{3n} \left[16+(4-4)^2 + 4\left[16+(4-(4+\Delta x))^2\right] + 2\left[16+(4-(4+2\Delta x))^2\right] + \cdots + 16+(4-14)^2\right] \]
04

Sum the results from both integrals

Add both integral approximations: \[ \int_0^{14}v(t) \, dt \approx \frac{4}{3n} \left[4(0) + 4\cdot4(\Delta x) + 2\cdot4(2\Delta x) + \cdots + 4(4) \right] + \frac{10}{3n} \left[16+(4-4)^2 + 4\left[16+(4-(4+\Delta x))^2\right] + 2\left[16+(4-(4+2\Delta x))^2\right] + \cdots + 16+(4-14)^2\right] \]
05

Calculate work done

Now we need to multiply the result by the constant force of 200 N to get the work done: \[ W = 200 \cdot \int_0^{14} v(t) dt \] Substituting the Simpson's rule approximation: \[ W \approx 200 \cdot \left[ \frac{4}{3n} \left(...\right) + \frac{10}{3n} \left(...\right) \right] \] With a desired number of intervals (n), we can calculate the work done. The exact value of the work done will depend on the number of intervals we choose and might be different if more accurate integration methods are employed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Work Done in Physics
In physics, work is defined as the transfer of energy from one object to another. It occurs when a force is applied to an object, causing it to move. The amount of work done (\( W \) is mathematically expressed by the equation
\[ W = F \times d \]
where \( W \) is the work done, \( F \) is the constant force applied, and \( d \) is the distance moved in the direction of the force. In the context of the given problem, the force is specified as 200 Newtons. To find the work done by this force as the object moves over a given time, we must calculate the distance traveled using the velocity-time relationship. Since the velocity varies over the time, we apply numerical integration to estimate the total distance traveled, then multiply by the force to calculate the work done.
Velocity-Time Relationship
Understanding the velocity-time relationship is essential to solving problems involving motion. Velocity is the rate of change of position with time; it tells us how fast an object is moving and in what direction. When graphed, the area under the velocity-time curve represents the distance traveled.
In our exercise, we are given two equations for velocity as functions of time (\(v(t)\)). The first equation applies from \(t = 0\) to \(t = 4\) seconds, and the second from \(t = 4\) to \(t = 14\) seconds. By integrating these velocity functions over the given time intervals, we calculate the total distance the object moves, which is needed to determine the work done.
Numerical Integration
Numerical integration is a cornerstone of computational mathematics, used when an integral cannot be evaluated analytically or when the function is only given by its values at certain points. The idea is to approximate the integral by summing up the areas of shapes (like rectangles or trapezoids) that closely 'fit' under the curve of the function being integrated.
Typically, the more subdivisions used in the numerical method, the more accurate the estimation becomes. In this exercise, numerical integration enables us to approximate the area under the velocity-time curve, thus estimating the distance traveled by the object subject to a varying velocity.
Simpson's Rule Application
Simpson's rule is a method for numerical integration that provides a more accurate approximation than other basic methods like the trapezoidal rule. It works by approximating the function to be integrated with parabolas that pass through multiple points on the curve. The rule applies the average of the method's estimates over several intervals, which significantly improves the accuracy of the integral estimation.
The rule's formula, as applied in this exercise, combines the function values at evenly spaced intervals with different weight factors, such as 1 for the endpoints, 4 for the odd-indexed points, and 2 for the even-indexed ones (excluding the first and last).
By applying Simpson's rule, we can approximate the integral of the velocity functions across different segments of time and thus calculate the work done on an object when a force is applied over time. The accuracy of Simpson's rule makes it a powerful tool for many physics applications, especially when coupled with a sensible choice of the number of intervals.

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