Problem 1 - Fundamentals of Physics

Question

What are (a) the \(x\) component and (b) the \(y\) component of a vector \(\vec{a}\) in the \(x y\) plane if its direction is \(250^{\circ} \mathrm{y}\) counterclockwise from the positive direction of the \(x\) axis and its magnitude is \(7.3 \mathrm{~m}\) ?

Step-by-Step Solution

Answer

The x component is approximately -2.5 m, and the y component is approximately -6.9 m.

Step 1: Understand the Problem

We are given a vector \(\vec{a}\) in the \(xy\)-plane with a direction of \(250^\circ\) counterclockwise from the positive \(x\)-axis and a magnitude of \(7.3\, \mathrm{m}\). Our task is to find the \(x\) and \(y\) components of this vector.

Step 2: Convert the Angle

The angle is given as \(250^\circ\) counterclockwise. To find the correct components, we remember that angles are measured counterclockwise from the positive \(x\)-axis, which is the standard reference direction.

Step 3: Use the Cosine and Sine for Components

The \(x\) component \(a_x\) of a vector can be found using \(a_x = a \cdot \cos(\theta)\), where \(\theta\) is the angle, and \(a\) is the magnitude of the vector. The \(y\) component \(a_y\) is given by \(a_y = a \cdot \sin(\theta)\). Since \(\theta = 250^\circ\):

Step 4: Calculate the X Component

Calculate the \(x\) component: \[a_x = 7.3 \cdot \cos(250^\circ).\]Using a calculator, we find \[a_x \approx 7.3 \cdot (-0.3420) \approx -2.5.\]

Step 5: Calculate the Y Component

Calculate the \(y\) component:\[a_y = 7.3 \cdot \sin(250^\circ).\]Using a calculator, we find \[a_y \approx 7.3 \cdot (-0.9397) \approx -6.9.\]

Step 6: Review the Results

The \(x\) component of vector \(\vec{a}\) is approximately \(-2.5\, \mathrm{m}\), and the \(y\) component is approximately \(-6.9\, \mathrm{m}\).

Key Concepts

Trigonometry

Trigonometry is essential to solving problems involving vector components in physics and mathematics. It helps us to convert between different forms of a vector. A vector in the plane can have various presentations, including its magnitude and direction (angle) or its components along the axes (here, x and y).
To retrieve useful information like the components of a vector from its magnitude and direction, we use trigonometric functions:

• Cosine (\(\cos\)) is used to find the horizontal component or the component along the x-axis.
• Sine (\(\sin\)) is used to determine the vertical component or the component along the y-axis.

By applying these functions, we can deconstruct a vector into more manageable parts to simplify many vector calculations.

Angle Conversion

Angle conversion is vital when working with vectors, especially if the angle differs from the usual reference points. A typical angle measure in these problems is degrees, defined relative to the positive x-axis. In this exercise, the vector's direction was given as \(250^{\circ}\) counterclockwise (CCW) from the positive x-axis.
When presented with such information:

• Remember that a full circle is \(360^{\circ}\). An angle over \(360^{\circ}\) means a complete rotation plus the additional degree specified.
• Angles can seamlessly convert to radians if necessary (not required here), using the conversion \(\frac{\pi}{180}\) for precision in calculations when needed.

Handling angles properly ensures the trigonometric values you use correspond correctly to the vector components.

XY Plane

The xy plane is a fundamental concept in geometry, representing a flat, two-dimensional surface with two axes:

• The x-axis (horizontal)
• The y-axis (vertical)

It provides a reference framework for defining vector directions and magnitudes. Generally, vector problems specify directions relative to these axes. In our exercise, knowing the vector's direction as \(250^{\circ}\) counterclockwise from the positive x-axis allows us to calculate components precisely.
This plane not only helps visualize vectors' positions but also serves as a reference for trigonometric calculations necessary to derive the components, ensuring they fit the plane's coordinate system.

Magnitude Calculation

Magnitude calculation involves determining the size or length of a vector. Although our exercise provided this (\(7.3 \mathrm{~m}\)), understanding how it integrates with direction allows component breakdown.

• The magnitude describes how far a vector reaches from the origin (0,0) in the xy plane.
• It always remains positive, as it signifies a distance or size.
• Trigonometry utilizes this with angle direction to solve for x and y components.

Given the magnitude, using its combination with cosine and sine allows for calculating exact dimensional extensions in horizontal and vertical directions. This precise breakdown is crucial in many physics applications, helping solve problems spanning from simple mechanics to advanced kinematics.