A football player runs directly down the field for 35 m before turning to
the right at an angle of 25° from his original direction and running an
additional 15 m before getting tackled. What is the magnitude and direc-
tion of the runner’s total displacement?
To find the magnitude and direction of the runner's total displacement, we can break it down into horizontal and vertical components.
Horizontal component:
The 35 m run down the field is in the horizontal direction, so its horizontal component is 35 m.
Vertical component:
The runner turns to the right at an angle of 25°. We can consider this as a right triangle, where the 35 m side is the adjacent side and the unknown displacement is the hypotenuse. Using trigonometry, we can find the vertical component.
cos(25°) = adjacent / hypotenuse
cos(25°) = 35 m / hypotenuse
hypotenuse = 35 m / cos(25°)
hypotenuse ≈ 38.14 m
Now we can find the vertical component:
Vertical component = hypotenuse * sin(25°)
Vertical component = 38.14 m * sin(25°)
Vertical component ≈ 16.32 m
Total displacement:
To find the total displacement, we can use the Pythagorean theorem since the horizontal and vertical components are perpendicular to each other.
Total displacement = √(horizontal component^2 + vertical component^2)
Total displacement = √((35 m)^2 + (16.32 m)^2)
Total displacement ≈ √(1225 m^2 + 266.3424 m^2)
Total displacement ≈ √(1491.3424 m^2)
Total displacement ≈ 38.62 m
The magnitude of the runner's total displacement is approximately 38.62 m.
Direction:
To find the direction, we can use trigonometry again. The tangent of the angle is equal to the vertical component divided by the horizontal component.
tan(θ) = vertical component / horizontal component
tan(θ) = 16.32 m / 35 m
θ = arctan(16.32 m / 35 m)
θ ≈ 25.831°
The direction of the runner's total displacement is approximately 25.831° from his original direction.
To solve this problem, we can break down the displacement into its horizontal and vertical components.
1. Start by finding the horizontal displacement:
- The initial horizontal displacement is 35 m, and after making a right turn, there is no further horizontal displacement.
- Therefore, the horizontal displacement is 35 m.
2. Next, find the vertical displacement:
- The initial vertical displacement is 0 m, as the player runs directly down the field.
- After making a right turn, there is no change in vertical displacement.
- Therefore, the vertical displacement is 0 m.
3. Calculate the magnitude of the total displacement:
- The total displacement is the hypotenuse of a right triangle formed by the horizontal and vertical displacements.
- Using the Pythagorean theorem, we can calculate the magnitude of the total displacement.
- Total displacement = sqrt(horizontal displacement^2 + vertical displacement^2)
- Total displacement = sqrt(35^2 + 0^2)
- Total displacement = sqrt(1225)
- Total displacement ≈ 35 meters
4. Calculate the direction of the total displacement:
- The direction can be determined using trigonometry.
- The angle after making the right turn is 25 degrees.
- The direction is measured relative to the initial direction, which was straight down.
- Therefore, the direction of the total displacement is 25 degrees to the right or clockwise from straight down.
So, the magnitude of the runner's total displacement is approximately 35 meters, and the direction of the total displacement is 25 degrees to the right or clockwise from straight down.