Question

A football field's length is exactly 100 yards, and its width is \(53 \frac{1}{3}\) yards. A quarterback stands at the exact center of the field and throws a pass to a receiver standing at one corner of the field. Let the origin of coordinates be at the center of the football field and the \(x\) -axis point along the longer side of the field, with the \(y\) -direction parallel to the shorter side of the field. a) Write the direction and length of a vector pointing from the quarterback to the receiver. b) Consider the other three possibilities for the location of the receiver at corners of the field. Repeat part (a) for each.

Short answer

Answer: The distance and the angle of the quarterback to each receiver are: a) Upper right corner: (57.735 yards, 27.964°) b) Upper left corner: (57.735 yards, 152.036°) c) Lower right corner: (57.735 yards, 332.036°) d) Lower left corner: (57.735 yards, 207.964°)

Step-by-step solution

Determine the dimensions of the field

We are given the length and width of the football field as 100 yards and \(53\frac{1}{3}\) (53.333) yards, respectively.

Represent the four corners of the field in the coordinate system

Since the origin is at the center of the football field, the coordinates for the four corners are: 1. Upper right corner: (50, 26.666) 2. Upper left corner: (-50, 26.666) 3. Lower right corner: (50, -26.666) 4. Lower left corner: (-50, -26.666)

Calculate the length and direction of a vector pointing from the quarterback to each receiver

First, we will use the Pythagorean theorem to find the distance from the origin (the quarterback's position) to each corner (the receiver's position). Then, we will use the arctangent function to find the angle. 1. Upper right corner: Distance: \(d_1 = \sqrt{(50)^2 + (26.666)^2} = 57.735\) yards Angle: \(\theta_1 = \arctan(\frac{26.666}{50}) = 27.964^\circ\) Vector: \((57.735, 27.964^\circ)\) 2. Upper left corner: Distance: \(d_2 = \sqrt{(-50)^2 + (26.666)^2} = 57.735\) yards Angle: \(\theta_2 = \arctan(\frac{26.666}{-50}) = 152.036^\circ\) Vector: \((57.735, 152.036^\circ)\) 3. Lower right corner: Distance: \(d_3 = \sqrt{(50)^2 + (-26.666)^2} = 57.735\) yards Angle: \(\theta_3 = \arctan(\frac{-26.666}{50}) = 332.036^\circ\) Vector: \((57.735, 332.036^\circ)\) 4. Lower left corner: Distance: \(d_4 = \sqrt{(-50)^2 + (-26.666)^2} = 57.735\) yards Angle: \(\theta_4 = \arctan(\frac{-26.666}{-50}) = 207.964^\circ\) Vector: \((57.735, 207.964^\circ)\) So, the vectors pointing from the quarterback to the receivers at each corner of the field are: a) Upper right corner: \((57.735, 27.964^\circ)\) b) Upper left corner: \((57.735, 152.036^\circ)\) c) Lower right corner: \((57.735, 332.036^\circ)\) d) Lower left corner: \((57.735, 207.964^\circ)\)

Key concepts

Coordinate System

Understanding the coordinate system is the foundation of vector calculation in physics. A coordinate system allows us to precisely locate a point in space by defining its position with numbers along axes. In a two-dimensional plane, we commonly use a Cartesian coordinate system, which includes two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where these axes intersect is known as the origin.

In the context of our football field problem, the origin is located at the center of the field, allowing us to define the position of the corners in relation to the center. By using this method, the location of each corner is given by a pair of numbers, or coordinates, which represent distances along the x and y axes. The coordinates enable us to create vectors that indicate both the position and the direction from the quarterback to the receiver.

Pythagorean Theorem

To calculate the length of the vector from the quarterback to each receiver, we first need to understand the Pythagorean theorem. This mathematical principle is incredibly useful in physics for determining the distance between two points in a plane. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In our exercise, we apply the theorem to find the distance from the origin to the corners of the field, considering the x and y values of the corner points as the two shorter sides of a right triangle. The length of the vector, representing the hypotenuse, is calculated using the formula: \(d = \sqrt{x^2 + y^2}\). This is an essential step in determining the magnitude of the displacement from the quarterback to the receiver.

Arctangent Function

The arctangent function, often abbreviated as \(\arctan\), is a trigonometric function that is fundamental in determining the direction of a vector. This function allows us to calculate the angle that a vector makes with the positive x-axis, based on the ratio of its y-component to its x-component. In practice, the angle helps us understand the vector's orientation in the plane.

By applying the arctangent function to the coordinates of the football field corners, we can find the angles mentioned in the exercise steps. For the upper right corner, for example, the angle \(\theta_1 = \arctan(\frac{26.666}{50})\) gives us the direction of the vector in degrees. This angle is important because it provides a complete description of the vector's orientation, which, combined with the magnitude obtained from the Pythagorean theorem, fully defines the vector from the quarterback to the receiver.