The football player: Vectors question

[alimf92]

1. A football player runs directly down the field for 35 m before turning to the right at an angle of 25 degrees from his original direction and running an additional 15 m before getting tackled. What is the magnitude and direction of the runner's total displacement?

Homework Equations

Pythagorean Theorem (c^2 = a^2 + b^2)
Sin, Cosine, and Tangent Functions
Inverse of Sin, Cosine, and Tangent functions



3. I do not have nay idea how to approach this question. Vectors is an easy concept, but I probably do not understand the context of the question itself. According to the textbook, the answer is "49 m at 7.3 degrees to the right of down-field".

 

[rl.bhat]

Th angle between the vectors is 25 degrees. Using cosine rule find resultant displacement.
To find the direction, take the component of 25 degrees along the original direction and perpendicular to the original direction. Using the tangent function find the direction.

 

[kerimek]

[/b]

I can explain the approach to solving this question using the concepts of vectors and trigonometry. First, let's define the variables in the question:

- The initial displacement (or distance) traveled by the football player is 35 m.
- The angle at which the player turns is 25 degrees.
- The additional displacement after turning is 15 m.

To find the total displacement, we need to combine these two displacements using vector addition. This can be done by breaking down the initial displacement into its horizontal and vertical components. Using trigonometric functions, we can calculate the horizontal component to be 35 cos 25 and the vertical component to be 35 sin 25.

Next, we add the additional displacement of 15 m to the horizontal component, and use the Pythagorean theorem to find the magnitude of the total displacement:

c^2 = (35 cos 25 + 15)^2 + (35 sin 25)^2
c^2 = 1225 cos^2 25 + 900 + 1225 sin^2 25
c^2 = 1225 (cos^2 25 + sin^2 25) + 900
c^2 = 1225 + 900
c = √2125 = 49 m

To find the direction of the total displacement, we use inverse trigonometric functions to find the angle:

tan θ = (35 sin 25) / (35 cos 25 + 15)
θ = tan^-1 ((35 sin 25) / (35 cos 25 + 15))
θ = 7.3 degrees

Therefore, the player's total displacement is 49 m at 7.3 degrees to the right of down-field. This means that if we draw a line from the starting point to the ending point of the player's movement, the angle between this line and the down-field direction is 7.3 degrees. This can also be seen as the angle the player turned from their original direction.