Sam travels east on his kayak for a distance 1.00km.Then,he heads 60 degrees South of East for 0.75km and finally veers at 20 degrees north of east for 0.5km.At the end of his trip,where is Sam reletion to his starting piont?
1.00 @ E = (1.000,0.000)
0.75 @ E60°S = (0.650,-0.375)
0.50 @ E20°N = (0.470,0.342)
Add 'em up and you end up at (1.62,-0.33)
which is 1.65 @ E11.51°S
Maths solving
To determine where Sam is in relation to his starting point, we can break down his journey into its eastward and northward components.
First, Sam travels east for a distance of 1.00 km. This means his eastward displacement is 1.00 km.
Next, he heads 60 degrees south of east for 0.75 km. To determine the eastward and northward distances traveled, we need to find the components of this displacement. The eastward component can be calculated using the cosine of the angle and multiplying it by the distance traveled:
Eastward component = 0.75 km * cos(60 degrees) = 0.75 km * 0.5 = 0.375 km.
The northward component can be calculated using the sine of the angle and multiplying it by the distance traveled:
Northward component = 0.75 km * sin(60 degrees) = 0.75 km * 0.866 = 0.6495 km.
Finally, Sam veers at 20 degrees north of east for 0.5 km. Again, we need to find the components of this displacement. The eastward component can be calculated using the cosine of the angle and multiplying it by the distance traveled:
Eastward component = 0.5 km * cos(20 degrees) = 0.5 km * 0.9397 = 0.4698 km.
The northward component can be calculated using the sine of the angle and multiplying it by the distance traveled:
Northward component = 0.5 km * sin(20 degrees) = 0.5 km * 0.342 = 0.171 km.
To determine the final displacement, we sum up the eastward and northward components:
Final eastward displacement = 1.00 km + 0.375 km + 0.4698 km = 1.8448 km.
Final northward displacement = 0.6495 km + 0.171 km = 0.8205 km.
Therefore, Sam is located 1.8448 km east and 0.8205 km north from his starting point.
To determine Sam's final position relative to his starting point, we can break down each leg of his journey and calculate x and y coordinates.
First, Sam travels east for a distance of 1.00 km. This means his position changes by 1.00 km in the positive x direction, so his new position is (1.00, 0).
Next, Sam heads 60 degrees South of East for 0.75 km. To calculate the x and y coordinate changes, we need to find the components of this displacement. The horizontal component (in the x direction) can be found by multiplying the total distance by the cosine of the angle:
horizontal component = 0.75 km * cos(60 degrees) = 0.75 km * 0.5 = 0.375 km
The vertical component (in the y direction) can be found by multiplying the total distance by the sine of the angle:
vertical component = 0.75 km * sin(60 degrees) = 0.75 km * sqrt(3)/2 = 0.75 km * 0.866 = 0.6495 km
Adding these components to the previous position, we get:
x coordinate = 1.00 km + 0.375 km = 1.375 km
y coordinate = 0 + (-0.6495 km) = -0.6495 km
So his new position is approximately (1.375, -0.6495).
Finally, Sam veers 20 degrees North of East for 0.5 km. Again, calculate the x and y coordinate changes using the components. The horizontal component (in the x direction) can be found by multiplying the total distance by the cosine of the angle:
horizontal component = 0.5 km * cos(20 degrees) = 0.5 km * 0.9397 = 0.4698 km
The vertical component (in the y direction) can be found by multiplying the total distance by the sine of the angle:
vertical component = 0.5 km * sin(20 degrees) = 0.5 km * 0.3420 = 0.1710 km
Adding these components to the previous position, we get:
x coordinate = 1.375 km + 0.4698 km = 1.8448 km
y coordinate = -0.6495 km + 0.1710 km = -0.4785 km
So Sam's final position, relative to his starting point, is approximately (1.845 km, -0.479 km).
To summarize:
Starting point: (0, 0)
East leg: (1.00, 0)
South leg: (1.375, -0.6495)
Final leg: (1.845, -0.479)