a hiker walks 10.0km(NE), 5.0 (W) and then 2.0km(S) in 2.5h. What is the hikers displacement?
To find the hiker's displacement, we need to determine the straight-line distance and direction from the starting point to the final position.
Let's break down the given information:
1. The hiker walks 10.0 km in a northeast direction.
2. Then, the hiker walks 5.0 km in a west direction.
3. Lastly, the hiker walks 2.0 km in a south direction.
Let's start by finding the net north-south distance:
The hiker walked 10.0 km northeast (NE), which can be split into north and east components. To determine the north component, we use the sine function:
North component = 10.0 km * sin(45 degrees) ≈ 7.07 km
Next, the hiker walked 2.0 km south.
Net north-south distance = 7.07 km - 2.0 km = 5.07 km south
Now, let's find the net east-west distance:
The hiker walked 10.0 km northeast (NE), which can be split into north and east components. To determine the east component, we use the cosine function:
East component = 10.0 km * cos(45 degrees) ≈ 7.07 km
Next, the hiker walked 5.0 km west.
Net east-west distance = 7.07 km - 5.0 km = 2.07 km east
Finally, we can calculate the displacement using the Pythagorean theorem:
Displacement = √((net east-west distance)^2 + (net north-south distance)^2)
Displacement = √((2.07 km)^2 + (5.07 km)^2) ≈ √(4.2849 km^2 + 25.7049 km^2) ≈ √29.9898 km^2 ≈ 5.48 km
Therefore, the hiker's displacement is approximately 5.48 km, and it is in a southeasterly direction.
To find the hiker's displacement, we can break down the distances and directions of the hiker's movements into their vector components.
First, let's represent the directions using a coordinate system. We can use the positive x-axis for the east direction (NE), the negative y-axis for the west direction (W), and the negative x-axis for the south direction (S). This way, any movement to the right (east) will have a positive x-component, movement to the left (west) will have a negative x-component, and movement downwards (south) will have a negative y-component.
Given:
Distance moved in the NE direction = 10.0 km
Distance moved in the W direction = 5.0 km
Distance moved in the S direction = 2.0 km
Now, let's calculate the displacement along the x-axis and y-axis separately.
Displacement along the x-axis:
The hiker moved 10.0 km in the NE direction, which has an angle of 45 degrees with the positive x-axis.
Using trigonometry, we can find the x-component of this movement: cos(45 degrees) * 10.0 km = 7.07 km (rounded to 2 decimal places).
The hiker moved 5.0 km in the W direction, which has an angle of 180 degrees with the positive x-axis. The x-component of this movement will be: cos(180 degrees) * (-5.0 km) = -5.0 km.
Adding these two x-components: 7.07 km + (-5.0 km) = 2.07 km (rounded to 2 decimal places). So the displacement along the x-axis is 2.07 km.
Displacement along the y-axis:
The hiker moved 2.0 km in the S direction, which has an angle of 270 degrees with the positive x-axis.
Using trigonometry, we can find the y-component of this movement: sin(270 degrees) * (-2.0 km) = 2.0 km.
Adding these y-components: 2.0 km.
So the displacement along the y-axis is 2.0 km.
Therefore, the hiker's displacement is (2.07 km, 2.0 km).
The displacement is the distance and direction from the starting point. Add the vectors to see where the hiker ends up. The fact that it took 2.5 hours does not affect the answer.
If you do not know how to add vectors, there are a number of good sites where you can learn about that.
Using i-j notation (with j being a unit vector north and i a unit vector east) , the sum is
7.07 i + 7.07 j -5.00 i -2.00 j
= 2.07 i - 5.07 j
The magnitude of the displacement is 5.48 km. The direction is 22.2 degrees (arctan 0.408) south of east