A car is driven east for a distance of 44 km, then north for 25 km, and then in a direction 27° east of north for 28 km. Determine (a) the magnitude (in km) of the car's total displacement from its starting point and (b) the angle (from east) of the car's total displacement measured from its starting direction.
Add the three displacement vectors.
44i + 25j + 28sin27i + 28cos27j
= (44+28sin27)i + (25 +cos27)j
i and j are unit vectors east and north
Use the Pythagoreaqn theorem for the magnitude and the ratio of components for the tangent of the direction.
To determine the magnitude of the car's total displacement, we can use the Pythagorean theorem.
Step 1: Represent the car's motion on a Cartesian coordinate system. Assume the starting point is the origin (0,0) and assign the east direction as the positive x-axis, and the north direction as the positive y-axis.
Step 2: Calculate the displacement in the x-direction.
The car is driven east for a distance of 44 km, so its displacement in the x-direction is 44 km.
Step 3: Calculate the displacement in the y-direction.
The car is driven north for a distance of 25 km, so its displacement in the y-direction is 25 km.
Step 4: Calculate the displacement due to the component in the direction 27° east of north.
To find the component of the displacement in the direction 27° east of north, we need to find its x and y-components. Let's call them Dx and Dy, respectively.
Dx = displacement * cos(angle)
Dy = displacement * sin(angle)
Here, the displacement is 28 km and the angle is 27°.
Dx = 28 km * cos(27°)
Dx ≈ 24.804 km
Dy = 28 km * sin(27°)
Dy ≈ 12.863 km
Step 5: Calculate the total displacement in the x and y-directions.
The total displacement in the x-direction is the sum of the x-components:
Total displacement in x = 44 km + Dx ≈ 68.804 km
The total displacement in the y-direction is the sum of the y-components:
Total displacement in y = 25 km + Dy ≈ 37.863 km
Step 6: Calculate the magnitude of the total displacement using the Pythagorean theorem.
The magnitude of the total displacement is given by:
Magnitude = √(Total displacement in x)^2 + (Total displacement in y)^2
Magnitude ≈ √(68.804 km)^2 + (37.863 km)^2
Magnitude ≈ √(4735.453 km^2 + 1433.173 km^2)
Magnitude ≈ √6168.626 km^2
Magnitude ≈ 78.531 km
Therefore, the magnitude of the car's total displacement from its starting point is approximately 78.531 km.
Next, to determine the angle of the car's total displacement measured from its starting direction:
Step 1: Calculate the angle using the inverse tangent function.
Angle = arctan(Total displacement in y / Total displacement in x)
Angle = arctan(37.863 km / 68.804 km)
Angle ≈ arctan(0.5518)
Angle ≈ 29.755°
Therefore, the angle of the car's total displacement measured from its starting direction (east) is approximately 29.755°.