On a safari, a team of naturalists sets out toward a research station located 5.54 km away in a direction 47.9 ° north of east. After traveling in a straight line for 1.29 km, they stop and discover that they have been traveling 24.6 ° north of east, because their guide misread his compass. What are (a) the magnitude and (b) the direction (as a positive angle relative to due east) of the displacement vector now required to bring the team to the research station?
idk lol
Well, it seems like those naturalists are taking quite the scenic route! Let me calculate the magnitude and direction of their required displacement vector for you.
To find the magnitude, we can use the law of cosines. The initial distance they traveled (1.29 km) and the current distance to the research station (5.54 km) form a triangle. The included angle is the difference in the misread angles (24.6° - 47.9° = -23.3°). So, the equation looks like this:
c² = a² + b² - 2ab*cos(C)
Where c is the magnitude of the required displacement vector, a is 1.29 km, b is 5.54 km, and C is -23.3°.
Now, let me perform some calculations... *beep boop beep*
After crunching the numbers, I found that the magnitude of the required displacement vector is approximately 4.82 km. So, that's how far they need to go!
Now let's determine the direction. Since they initially wanted to go 47.9° north of east, we need to subtract the misread angle of 24.6° north of east from that. This gives us approximately 23.3° north of east.
So, the required displacement vector is approximately 4.82 km in magnitude and directed 23.3° north of east.
I hope this helps! Just remember, even if they're going in circles, at least they're having an adventure!
To find the magnitude and direction of the displacement vector required to bring the team to the research station, we can break down the problem into two separate displacements: the displacement already covered correctly, and the remaining displacement needed to reach the research station.
Let's start by calculating the displacement vector already covered correctly:
1. Calculate the x-component of the displacement already covered by using the given distance (1.29 km) and the angle north of east (24.6°):
x-component = distance × cos(angle)
= 1.29 km × cos(24.6°)
2. Calculate the y-component of the displacement already covered by using the same distance and angle:
y-component = distance × sin(angle)
= 1.29 km × sin(24.6°)
Next, let's calculate the remaining displacement:
1. Calculate the x-component of the remaining displacement by subtracting the x-component of the displacement already covered from the x-component required to reach the research station:
x-component = (5.54 km × cos(47.9°)) - x-component already covered
2. Calculate the y-component of the remaining displacement by subtracting the y-component already covered from the y-component required to reach the research station:
y-component = (5.54 km × sin(47.9°)) - y-component already covered
Now, we can find the magnitude and direction of the overall displacement vector:
1. Calculate the magnitude of the displacement vector using the Pythagorean theorem:
magnitude = √(x-component^2 + y-component^2)
2. Calculate the direction of the displacement vector using the inverse tangent function (arctan) to find the angle relative to due east:
direction = arctan(y-component / x-component)
Note: Be careful with the signs and quadrants when using the inverse tangent function. Also, consider adjusting the calculated angle if it falls in a different quadrant.
By following these steps, you can find the answers to both parts (magnitude and direction) of the displacement vector required to bring the team to the research station.
D = 5.54km[47.9o] - 1.29km[24.60]
X = 5.54*cos47.9 - 1.29*cos24.6=2.54 km.
Y = 5.54*sin47.9 - 1.29*sin24.6=3.57 km.
Tan A = Y/X = 3.57/2.54 = 1.40691
A = 54.6o
Displacement = Y/sin A = 3.57/sin54.6 =
4.38km[54.6o].
a. Magnitude = 4.38 km.
b. Direction = 54.6o N of E.