A car is driven east for a distance of 45 km, then north for 33 km, and then in a direction 32° east of north for 22 km. Determine the total displacement of the car from its starting point (counterclockwise from east).
Please show your own effort at solving this. It is a vector addition problem. We will be glad to critique your work.
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Superimpose an xyz coordinate system, such as this: x extends eastward, y extends northward. Then north is positive, west is negative, and south is negative. Add the three vectors, using the component technique.Draw the three vectors and the components of their net vector.
To determine the total displacement of the car, we need to break down each leg of the journey into its east and north components.
Leg 1: The car is driven east for a distance of 45 km.
The east component is 45 km, and the north component is 0 km since the car did not move north.
Leg 2: The car is driven north for 33 km.
The east component is 0 km since the car did not move east, and the north component is 33 km.
Leg 3: The car is driven in a direction 32° east of north for 22 km.
To find the east and north components, we need to determine the east and north distances.
The east distance is 22 km multiplied by the cosine of the angle 32° east of north.
The north distance is 22 km multiplied by the sine of the angle 32° east of north.
East distance = 22 km * cos(32°)
North distance = 22 km * sin(32°)
Now, we can calculate these values:
East distance = 22 km * cos(32°) ≈ 18.632 km
North distance = 22 km * sin(32°) ≈ 11.623 km
Next, we need to add up the east and north components of each leg to get the final displacement.
Total east displacement = 45 km + 0 km + 18.632 km = 63.632 km (counterclockwise from east)
Total north displacement = 0 km + 33 km + 11.623 km = 44.623 km
The total displacement of the car from its starting point can be found using the Pythagorean theorem:
Total displacement = √(Total east displacement^2 + Total north displacement^2)
Total displacement = √(63.632^2 + 44.623^2)
Total displacement ≈ √(4039.942 + 1990.999)
Total displacement ≈ √(6029.941)
Total displacement ≈ 77.6 km
Therefore, the total displacement of the car from its starting point (counterclockwise from east) is approximately 77.6 km.