Problem 37 - College Physics

Question

Two displacements, one with a magnitude of \(15.0 \mathrm{~m}\) and a second with a magnitude of \(20.0 \mathrm{~m},\) can have any angle you want. (a) How would you create the sum of these two vectors so it has the largest magnitude possible? What is that magnitude? (b) How would you orient them so the magnitude of the sum was at its minimum? What value would that be? (c) Generalize the result to any two vectors.

Step-by-Step Solution

Answer

(a) Align them in the same direction: 35 m. (b) Align them in opposite directions: 5 m. (c) Max is \(A+B\); min is \(|A-B|\).

Step 1: Understanding Vector Addition

Recall that the sum of two vectors depends on the angle between them. The magnitude of the resultant vector can be calculated using the formula for vector addition: \ \[ R = \sqrt{A^2 + B^2 + 2AB\cos(\theta)} \] \ where \(R\) is the resultant vector, \(A\) and \(B\) are the magnitudes of the two vectors, and \(\theta\) is the angle between them.

Step 2: Calculate Maximum Magnitude

To maximize the magnitude of the resultant vector, the two vectors should be aligned in the same direction. This occurs when \(\theta = 0\), and thus \(\cos(0) = 1\). Substitute \(\theta = 0\) into the formula: \ \[ R_{max} = \sqrt{15^2 + 20^2 + 2 \times 15 \times 20 \times 1} = 15 + 20 = 35 \; \mathrm{m} \] \ The maximum magnitude is 35 m.

Step 3: Calculate Minimum Magnitude

To minimize the magnitude of the resultant vector, the two vectors should be in opposite directions. This occurs when \(\theta = 180^{\circ}\), and \(\cos(180^{\circ}) = -1\). Substitute into the formula: \ \[ R_{min} = \sqrt{15^2 + 20^2 + 2 \times 15 \times 20 \times (-1)} = |20 - 15| = 5 \; \mathrm{m} \] \ The minimum magnitude is 5 m.

Step 4: Generalize to Any Two Vectors

For any two vectors of magnitudes \(A\) and \(B\), the maximum magnitude of their sum is \(A + B\), occurring when they are in the same direction (\(\theta = 0\)). The minimum magnitude of their sum is \(|A - B|\), occurring when they are in opposite directions (\(\theta = 180^{\circ}\)).

Key Concepts

Resultant Magnitude

When two vectors add up, the size or length of their sum is called the resultant magnitude. To understand this better, imagine the two vectors as arrows pointing from a starting point. The resultant vector, or the sum, is like a new arrow showing the total combined movement of both original vectors. The equation to find the resultant magnitude is:\[ R = \sqrt{A^2 + B^2 + 2AB\cos(\theta)} \]where:- \(R\) is the resultant magnitude.- \(A\) and \(B\) are the magnitudes of the original two vectors.- \(\theta\) is the angle between the two vectors.This equation tells us that the resultant depends on both the sizes and directions of the individual vectors. By understanding this concept, we can find out how two movements combine to give a total result.

Angle Between Vectors

The angle between vectors significantly affects their resultant magnitude. Two important cases illustrate this: - **Maximum Resultant**: When the vectors are in the same direction, the angle \(\theta\) is \(0\) degrees. This makes \(\cos(0)=1\), leading to the largest possible resultant magnitude, calculated as the simple sum of the two vectors: \(A + B\).- **Minimum Resultant**: When the vectors point in opposite directions, the angle is \(180\) degrees, where \(\cos(180)=-1\). The resultant in this arrangement becomes the difference between the two magnitudes, \(|A - B|\).These scenarios show how aligning vectors can either maximize or minimize the total displacement.

Vector Components

Vectors can be broken down into smaller parts called components. These are typically along the horizontal (x-axis) and vertical (y-axis) directions. Components are useful because they allow us to simplify complex vector addition into 1-dimensional problems:- **Horizontal Component**: Given by \(A_x = A \cos(\theta)\), where \(A_x\) is the x-component.- **Vertical Component**: Given by \(A_y = A \sin(\theta)\), where \(A_y\) is the y-component.These components are crucial for calculating actual placement or displacement of vectors in a coordinate system.

Vector Direction

The direction of a vector is often expressed as the angle it makes with a reference line, such as the positive x-axis in a coordinate plane. Direction is essential because it specifies the path along which the vector acts. Consider:

• Vectors in the same direction (angle \(\theta\) of \(0\) degrees) reinforce each other, leading to a larger resultant.
• Vectors at 90 degrees create a resultant at an angle, combining perpendicular components.
• Vectors in opposite directions (angle \(180\) degrees) counteract each other, reducing the resultant magnitude.

Understanding vector direction is important for predicting how vectors behave and interact.