Problem 14 - APEX Calculus

Question

Exercises \(13-16\) ask a variety of questions dealing with approximating error and sensitivity analysis. Projectile Motion: The \(x\) -value of an object moving under the principles of projectile motion is \(x\left(\theta, v_{0}, t\right)=\) \(\left(v_{0} \cos \theta\right) t\). A particular projectile is fired with an initial velocity of \(v_{0}=250 \mathrm{ft} / \mathrm{s}\) and an angle of elevation of \(\theta=60^{\circ}\). It travels a distance of \(375 \mathrm{ft}\) in 3 seconds.

Step-by-Step Solution

Answer

The calculations confirm the projectile traveled 375 ft in 3 seconds.

Step 1: Understand the Formula

The projectile's x-position is given by \( x(\theta, v_{0}, t) = (v_{0} \cos \theta) t \). This shows how the distance depends on the initial velocity \( v_{0} \), the angle \( \theta \), and the time \( t \).

Step 2: Substitute Known Values

Substitute \( v_{0} = 250 \, \text{ft/s} \), \( \theta = 60^{\circ} \), and \( t = 3 \, \text{s} \) into the equation: \[ x = (250 \cos 60^{\circ}) \times 3. \]

Step 3: Calculate \( \cos 60^{\circ} \)

The cosine of 60 degrees is \( \cos 60^{\circ} = \frac{1}{2} \).

Step 4: Simplify the Expression

Substitute \( \cos 60^{\circ} = \frac{1}{2} \) into the equation: \[ x = (250 \times \frac{1}{2}) \times 3. \] This simplifies to \( x = 125 \times 3 \).

Step 5: Final Calculation

Multiply the values to find the x-distance: \( x = 375 \, \text{ft} \).

Step 6: Verify the Calculation

Confirm that the calculated distance of \( 375 \, \text{ft} \) matches the distance mentioned in the problem statement.

Key Concepts

Error Approximation

In projectile motion, it's important to ensure your calculations are accurate by approximating errors. In this exercise, the x-position of the projectile was determined by the formula: \[ x(\theta, v_{0}, t) = (v_{0} \cos \theta) t \]

1. Approximating the error begins with substituting the known values such as \( v_{0} = 250 \, \mathrm{ft/s} \), \( \theta = 60^{\circ} \), and \( t = 3 \, \text{s} \).
2. Then, calculate \( \cos 60^{\circ} \) which is \( \frac{1}{2} \).
3. Plug in these values into the equation to find the x-position.

If the calculated x-distance does not match with the one given (375 ft, in this case), you need to consider potential sources of error like rounding errors in trigonometric calculations or incorrect substitution of values. Also, checking the consistency of units and computed angles could be vital in approximating and minimizing errors.Take note, even small numerical inaccuracies can lead to significant shifts in projectile path predictions.

Sensitivity Analysis

Sensitivity analysis explores how changes in variables affect the outcome, which in this case is the projectile's x-distance. Let us consider the impacts of slightly varying each parameter:- **Angle \(\theta\):** A small change in the angle can significantly alter \( \cos(\theta) \) and, consequently, the x-position. Factors such as instrument precision may affect the angle readings.- **Initial velocity \(v_{0}\):** If \(v_{0}\) changes, the projectile covers more or less distance. For example, increasing \(v_{0}\) slightly while keeping angle and time constant results in a corresponding increase in x-distance.- **Time \(t\):** Variation in time, perhaps due to measurement error or other disturbances, impacts the range the projectile achieves.To conduct sensitivity analysis, one manually adjusts each variable while keeping others constant to observe alterations in x-distance. This method helps determine the extent each factor contributes to the overall position. Understanding these sensitivities is crucial for precision in fields like engineering and physics, where control over motion behavior is necessary.

Trigonometric Functions

Trigonometric functions play a crucial role in analyzing projectile motion. The function \(\cos(\theta)\) is used here to determine the horizontal component of the initial velocity. Remember the key steps involved:

• The function \(\cos(\theta)\) is essential to separate the components of projectile velocity into horizontal (x-axis) and vertical (y-axis) avenues.
• For \(\theta = 60^{\circ}\), finding \(\cos(60^{\circ})\) helps in simplifying the mathematical expression: \( \cos(60^{\circ}) = \frac{1}{2} \).

The importance of trigonometric functions like cosine extends beyond just solving this projectile problem; they apply to any analysis involving angles and sides of triangles. **Cosine** in particular is powerful in determining how much of a vector aligns along a particular direction, fundamental in physics for breaking vectors into components. Developing a strong understanding of these functions is fundamental for anyone working with angles and motion, making them indispensable in academic and practical settings.