A ship travels 100 km due north on the first day of a voyage, 60 km north-east on the second day, and 120 km due east on the third day. Find the resultant displacement.
North:
100 + 60(.5 sqrt 2 ) = 142
East:
60(.5 sqrt 2) + 120 = 162
Tan angle east of north (compass course made good) = 162/142 = 1.14
so
angle east of north = 48.7 degrees
distance from start = sqrt(162^2+142^2)
To find the resultant displacement, we need to add up all the individual displacements of the ship.
On the first day, the ship travels 100 km due north. Since this is a straight line, the displacement is simply 100 km in the north direction.
On the second day, the ship travels 60 km north-east. To find the displacement, we can treat this as a right-angled triangle. The north-east direction is exactly between north and east, so the angle is 45 degrees. Using trigonometry, we can calculate the north and east components of this displacement. The north component = 60 km * cos(45°) = 60 km * √(2)/2 ≈ 42.4 km. The east component = 60 km * sin(45°) = 60 km * √(2)/2 ≈ 42.4 km.
On the third day, the ship travels 120 km due east. Since this is a straight line, the displacement is simply 120 km in the east direction.
Now, we can add up all the displacements to find the resultant displacement. The north component adds up to 100 km + 42.4 km = 142.4 km in the north direction. The east component adds up to 120 km + 42.4 km = 162.4 km in the east direction.
To find the resultant displacement, we can use the Pythagorean theorem. The magnitude of the resultant displacement is the square root of the sum of the squares of the components: √(142.4 km² + 162.4 km²) ≈ 212.8 km.
Therefore, the resultant displacement is approximately 212.8 km, and it is in the north-east direction.
To find the resultant displacement, we first need to break down the given displacements into their respective horizontal (x-axis) and vertical (y-axis) components.
On the first day, the ship travels 100 km due north. This can be represented as a displacement of (0, 100), where 0 km is the horizontal component and 100 km is the vertical component.
On the second day, the ship travels 60 km northeast. To find the horizontal and vertical components, we can use trigonometry. The angle between the north direction and northeast direction is 45 degrees (since it forms a right triangle). Let's call the horizontal component H and the vertical component V.
H = 60 * cos(45°) ≈ 42.4 km
V = 60 * sin(45°) ≈ 42.4 km
Therefore, the second-day displacement can be represented as (42.4, 42.4).
On the third day, the ship travels 120 km due east. This can be represented as a displacement of (120, 0), where 120 km is the horizontal component and 0 km is the vertical component.
To find the resultant displacement, we need to add the individual displacements together by adding their respective horizontal and vertical components.
Horizontal component = 0 + 42.4 + 120 = 162.4 km
Vertical component = 100 + 42.4 + 0 = 142.4 km
Therefore, the resultant displacement is approximately (162.4, 142.4) km.