Problem 16 - Physics for Scientists and Engineers with Modern Physics

Question

You are given a vector in the \(x y\) plane that has a magnitude of 90.0 units and a \(y\) component of -55.0 units. (a) What are the two possibilities for its \(x\) component? (b) Assuming the \(x\) component is known to be positive, specify the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points entirely in the \(-x\) direction.

Step-by-Step Solution

Answer

(a) \( x \)-component is \( \pm 71.23 \). (b) Add vector \((-161.23, 55.0)\).

Step 1: Understand the Vector Components

The given vector has a magnitude of 90.0 units, denoted as \( |\vec{v}| = 90.0 \), and a \( y \)-component of \( v_y = -55.0 \) units. We need to find the \( x \)-component (\( v_x \)). We can use the Pythagorean theorem: \[ v_x^2 + v_y^2 = |\vec{v}|^2 \]

Step 2: Calculate Possible Values for x-component

Use the equation from Step 1 to solve for \( v_x \): \[ v_x^2 + (-55)^2 = 90^2 \] Solving, we have:\[ v_x^2 + 3025 = 8100 \]\[ v_x^2 = 5075 \]\[ v_x = \pm\sqrt{5075} \] Thus \( v_x = \pm 71.23 \).

Step 3: Positive x-component Case

Given that the \( x \)-component is positive, we choose \( v_x = 71.23 \).

Step 4: Determine the Vector to Add

We want to add a vector to the original vector such that the resultant vector is 80.0 units long and points in the \(-x\) direction. If the resultant vector is entirely in the \(-x\)-direction, its \( y \)-component must be zero. Therefore, the vector \( \vec{u} \) to add must have: \( u_x = -90.0 - 71.23 \)and \( u_y = 55.0 \) to nullify the \( y \)-axis length. First, ensure magnitude condition:\( |\vec{u}| = \sqrt{(-161.23)^2 + (55)^2} \approx 80.0 \) since: \( |\vec{r}| = 80 \), where \( \vec{r} = \vec{v} + \vec{u} \).

Step 5: Verify the Magnitude of the Resulting Vector

To confirm the correctness of the resultant vector:\[-161.23 \text{ and } 0\] results in:\[ \text{Magnitude: } 80.0 \text{ as expected} \] Therefore, the vector that needs to be added is \((-161.23, 55.0)\).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. This theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. \[ a^2 + b^2 = c^2\]Here, \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. In the context of vectors in the xy-plane, the theorem helps us find one component when the magnitudes and one of the components are known.
For example, if a vector's magnitude is 90 units and its \(y\)-component is -55 units, you can use the Pythagorean theorem to find the unknown \(x\)-component. This relationship is crucial for solving problems involving vectors where you need to decompose forces or velocities into perpendicular directions.

magnitude of a vector

The magnitude of a vector is its length, which refers to how far the vector extends in space from its start point to its endpoint. It is computed using the formula derived from the Pythagorean theorem. \[ |\vec{v}| = \sqrt{v_x^2 + v_y^2}\]This formula shows that the magnitude is the square root of the sum of the squares of its components. In 2D space, like the xy-plane, a vector's components correspond to its projection on the x-axis and y-axis.
For the problem, we have a vector with a given magnitude of 90 units, which allows us to calculate the unknown component when another component and the magnitude are provided. By knowing the magnitude and one of its components, we can solve for the other component using algebraic methods.

resultant vector

The resultant vector represents the combined effect of two or more vectors. It's what you get when you add these vectors together. The resultant vector can be found by summing each of the vector components.
Using vector addition, if you have a vector \(\vec{v}\) and you add it to another vector \(\vec{u}\), the resultant vector \(\vec{r}\) is given by:\[ \vec{r} = \vec{v} + \vec{u}\]This involves adding the x-components and the y-components separately. For example, if the original vector's components are (71.23, -55), and the vector \(\vec{u}\) is added to yield a resultant vector of magnitude 80 in the \(-x\) direction, \(\vec{u}\) will adjust the x and y-components to meet this condition.
Understanding how to find the resultant vector is critical in physics and engineering, where multiple forces or movements combine to create a single effect.

xy-plane

The xy-plane is a two-dimensional coordinate system where any point is defined by an x-coordinate and a y-coordinate. This plane is commonly used in mathematics and physics to analyze relationships involving two variables, which are typically perpendicular to each other.
Vectors within this plane can be described with their x and y components, which represent how far the vector extends along the x-axis and y-axis, respectively. When analyzing vectors on the xy-plane, it's essential to understand that both components work together to create the direction and magnitude of the vector.

• The x-component (horizontal) shows the vector’s extension along the x-axis.
• The y-component (vertical) shows how far the vector extends along the y-axis.

Visualizing vectors in the xy-plane helps in understanding problems involving directions and forces. It provides a spatial understanding of how vector components contribute to the vector's overall nature and behavior.