In a game of American football, a quarterback takes the ball from the line of scrimmage and runs backwards for 10 yards, and then sideways to the left, parallel to the line of scrimmage for 15 yards. At this point, he throws a forward pass at the angle of 60 degrees with respect to the line of scrimmage to the right. Ball flies 50 yards in this direction, before it is caught. What is the total distance between were the ball was initially spotted and were it was caught?

a)Write down x and y components of each vector.
b)Add components up to obtain components of the total displacement.
c)Calculate magnitude of the total displacement vector.

a)

Let's break down the quarterback's movement into vector components.

1) Running Backwards: The quarterback runs backwards for 10 yards. This can be represented as a displacement vector with an x-component of -10 yards (since he is moving in the opposite direction of the line of scrimmage) and a y-component of 0 yards (since he is not moving sideways).

2) Sideways: The quarterback then moves sideways to the left for 15 yards. This can be represented as a displacement vector with an x-component of 0 yards (since he is not moving forwards or backwards) and a y-component of -15 yards (since he is moving to the left parallel to the line of scrimmage).

3) Forward Pass: The quarterback throws a forward pass at an angle of 60 degrees with respect to the line of scrimmage to the right direction for 50 yards. We need to find the x and y components of this forward pass vector.

The x-component can be found using trigonometry:
x-component = 50 yards * cos(60 degrees) = 50 yards * 0.5 = 25 yards

The y-component can also be found using trigonometry:
y-component = 50 yards * sin(60 degrees) = 50 yards * √3/2 ≈ 43.3 yards

So, the x-component of the forward pass vector is 25 yards, and the y-component is approximately 43.3 yards.

b)

To find the components of the total displacement, we add up the x and y components separately.

x-component of total displacement = -10 yards + 0 yards + 25 yards
= 15 yards

y-component of total displacement = 0 yards + (-15 yards) + 43.3 yards
= 28.3 yards

c)

To calculate the magnitude of the total displacement vector, we can use the Pythagorean theorem.

Magnitude of total displacement vector = √((x-component)^2 + (y-component)^2)
= √((15 yards)^2 + (28.3 yards)^2)
≈ √(225 yards^2 + 800.89 yards^2)
≈ √(1025.89 yards^2)
≈ 32.05 yards

Therefore, the total distance between where the ball was initially spotted and where it was caught is approximately 32.05 yards.