A hiker walks 10.0 km[NE], 5.0 km[W], and then 2.0 km[S] in 2.5 h.
(a) What is the hiker's displacement?
(b) In what direction must the hiker set out, in order to return by the most direct route to the starting point?
(c) If the hiker walks at a constant speed for the entire trip and returns by the most direct route, how long will the total walk take?
Thanks!
(a) Well, to find the displacement, we need to find the total distance traveled and the direction. The hiker walked 10.0 km[NE], 5.0 km[W], and 2.0 km[S]. To find the total distance, we can use the Pythagorean theorem. So, the total distance traveled is the square root of (10.0^2 + 5.0^2 + (-2.0)^2), which is approximately 11.1 km. As for the direction, we can add up the vectors to find the resultant vector. So, the hiker's displacement is approximately 11.1 km at an angle of 74.5°.
(b) To return to the starting point in the most direct route, the hiker should walk in the opposite direction of the displacement. So, the hiker should set out in the direction opposite to 74.5°. But hey, if you're feeling adventurous, you could always take a detour and explore some new trails!
(c) If the hiker walks at a constant speed for the entire trip and returns by the most direct route, we can calculate the time using the distance and the speed. Since the hiker walked approximately 11.1 km and we're not given the speed, it's impossible to determine the time accurately. But hey, don't worry too much about time when you're out exploring nature – just enjoy the journey, and the time will fly by!
To find the hiker's displacement, we need to calculate the resultant displacement vector by adding the individual displacements in the x and y directions.
Let's assume East as the positive x-direction and North as the positive y-direction.
(a) The displacement in the x-direction:
Since the hiker walks 10.0 km[NE] and 5.0 km[W], we can calculate the net displacement in the x-direction by subtracting the displacement to the west from the displacement to the northeast.
Displacement in the x-direction = 10.0 km cos(45°) - 5.0 km = 10.0 km / √2 - 5.0 km ≈ 3.54 km - 5.0 km ≈ -1.46 km
The displacement in the y-direction:
Since the hiker walks 10.0 km[NE] and then 2.0 km[S], the net displacement in the y-direction can be calculated by adding the displacement to the northeast and subtracting the displacement to the south.
Displacement in the y-direction = 10.0 km sin(45°) - 2.0 km = 10.0 km / √2 - 2.0 km ≈ 3.54 km - 2.0 km ≈ 1.54 km
Now we can calculate the resultant displacement using the Pythagorean theorem:
Resultant Displacement = √((Displacement in x-direction)^2 + (Displacement in y-direction)^2)
Resultant Displacement = √((-1.46 km)^2 + (1.54 km)^2)
Resultant Displacement ≈ √(2.1316 km^2 + 2.3716 km^2)
Resultant Displacement ≈ √4.5032 km^2
Resultant Displacement ≈ 2.12 km
Therefore, the hiker's displacement is approximately 2.12 km, and the direction of the displacement is towards the southeast.
(b) To return to the starting point by the most direct route, the hiker should set out in the direction opposite to their displacement. So, the hiker should set out in the northwesterly direction.
(c) If the hiker walks at a constant speed for the entire trip and returns by the most direct route, the total distance covered will be equal to the magnitude of the displacement, which is approximately 2.12 km. Given that the hiker's speed is constant, the time taken will be the total distance covered divided by the speed.
Total Time = Total Distance / Speed
Total Time = 2.12 km / (10.0 km/hour) [assuming the speed is 10.0 km/hour]
Total Time ≈ 0.212 hours ≈ 12.7 minutes.
Therefore, if the hiker walks at a constant speed and returns by the most direct route, the total walk will take approximately 12.7 minutes.
To solve this problem, we are going to use vector addition to find the hiker's displacement. The hiker's displacement refers to the straight-line distance and direction from the starting point to the ending point.
(a) To find the hiker's displacement:
Step 1: Break down the hiker's movements into individual vectors.
- The hiker walks 10.0 km[NE]. This can be represented as a vector moving 10.0 km in the northeast direction.
- The hiker walks 5.0 km[W]. This can be represented as a vector moving 5.0 km in the west direction.
- The hiker walks 2.0 km[S]. This can be represented as a vector moving 2.0 km in the south direction.
Step 2: Add up the individual vectors to find the resultant vector.
- Start by aligning the vectors along the same axes. In this case, let's align them along the north (N) and east (E) directions.
- Convert the direction of each vector to its corresponding component along the N and E axes. For example, the 10.0 km[NE] vector can be broken down into two components: 10.0 km along N and 10.0 km along E (since NE is a combination of N and E).
- Add up the components along the N and E axes separately to find the resultant vector.
Summing up the components:
Total displacement along the N axis: 10.0 km - 2.0 km = 8.0 km[S]
Total displacement along the E axis: 10.0 km - 5.0 km = 5.0 km[E]
The displacement vector can be written as (5.0 km[E], 8.0 km[S]).
Therefore, the hiker's displacement is 5.0 km east and 8.0 km south.
(b) To find the direction the hiker must set out to return by the most direct route, we need to determine the angle of the resultant vector.
- Use the tangent function to find the angle:
angle = arctan(8.0 km / 5.0 km) ≈ 57.99°
- The angle is measured counterclockwise from the positive east direction.
Therefore, the hiker must set out at an angle of approximately 57.99° north of east in order to return by the most direct route to the starting point.
(c) If the hiker returns by the most direct route, the total distance traveled will be equal to the magnitude of the displacement.
- Calculate the magnitude of the displacement using the Pythagorean theorem:
magnitude = sqrt((5.0 km)^2 + (8.0 km)^2) ≈ 9.43 km
- Divide the magnitude by the average speed of the hiker to find the total time:
total time = 9.43 km / (2.5 h) ≈ 3.77 h
Therefore, the total walk will take approximately 3.77 hours.
Let the starting point be in the origin of coordinate system. Then the points of the hicker’s route are A , B and C.
AB=10 => A (7.1; 7.1),
BC =5 => B(2.1; 7.1),
BC=2 => C (2.1; 5.1)
Displacement BC = sqrt(2.1²+5.1²)=5.22 km
The angle with +x-axis is arctan(5.1/2.1)=67.6º
s=10+5+2=17 km,
v=s/t=17/2.5 =6.8 km/h
L=10+5+2+5.22 =22.22 km,
t1=L/v=/6.8=3.27 h.