An archaeologist begins a trip by first walking 30km southeast from the campsite. He stops and sets up a temporal camp for the night. On the second day, he walks 60km in direction 60 degree north of east, at which point he discovers the ancient city of gold.
a) Determine the components of the archaeologist's displacement for each day
b) Determine the components of the archaeologist's displacement R for the trip. Find the expression for R in terms of unit vectors
To determine the components of the archaeologist's displacement for each day, we can break it down into its eastward (x-axis) and northward (y-axis) components:
Day 1:
- Displacement in the x-axis (eastward) = -30 km (because the archaeologist walked southeast)
- Displacement in the y-axis (northward) = 0 km (since there is no northward movement)
Day 2:
To determine the displacement in the x-axis and y-axis for day 2, we can convert the given direction 60 degrees north of east to its components using trigonometry:
- Displacement in the x-axis (eastward) = 60 km * cos(60°)
= 60 km * (1/2)
= 30 km
- Displacement in the y-axis (northward) = 60 km * sin(60°)
= 60 km * (√3/2)
= 30√3 km
b) To determine the total displacement R for the entire trip, we can add the displacement components from each day together.
Displacement in the x-axis (eastward) = (-30 km) + (30 km)
= 0 km
Displacement in the y-axis (northward) = 0 km + (30√3 km)
= 30√3 km
Therefore, the expression for the displacement vector R in terms of unit vectors i and j is:
R = 0 km * i + 30√3 km * j
or,
R = 0i + 30√3j
To find the components of the archaeologist's displacement, we can break down the distances and directions into their vector components.
a) Displacement for each day:
Day 1: Walking 30km southeast
- Southeast direction can be expressed as a vector (-1, -1) or (-1 km, -1 km) in terms of unit vectors.
- Therefore, the displacement for the first day is (-30 km, -30 km).
Day 2: Walking 60km in direction 60 degrees north of east
- To find the vector components, we need to split the given distance and direction into horizontal and vertical components.
- The horizontal component can be found using cos(60°) * 60km = 30 km.
- The vertical component can be found using sin(60°) * 60km = 51.96 km.
- Therefore, the displacement for the second day is (30 km, 51.96 km).
b) To find the components of the archaeologist's displacement R for the entire trip, we can add the displacements of each day together.
R = Day 1 Displacement + Day 2 Displacement
= (-30 km, -30 km) + (30 km, 51.96 km)
Now, we need to combine the components with the unit vectors.
Using the notation i and j to represent the unit vectors, where i represents horizontal motion and j represents vertical motion, the expression for R in terms of unit vectors is:
R = (-30 km, -30 km) + (30 km, 51.96 km)
= -30 km i - 30 km j + 30 km i + 51.96 km j
= (30 km i + 30 km i) + (-30 km j + 51.96 km j)
= 60 km i + 21.96 km j
Therefore, the expression for R in terms of unit vectors is R = 60 km i + 21.96 km j.