You drive due east at 86km/h for 55min . Then you follow a road going 60 degrees south of east, going 99km/h for another 40min .
Find your total displacement
Enter the x and y components of the displacement separated by a comma. Assume that the positive x and y axes are directed to the east and north, respectively.
Compute the distances travelled in both legs of the journey, and perform a vector addition.
can you show me how to work this out?
To find the total displacement, we need to calculate the horizontal and vertical components of displacement separately and then combine them.
First, let's calculate the horizontal displacement:
Horizontal displacement from the first leg of the journey:
Time = 55 minutes = 55/60 = 0.92 hours
Speed = 86 km/h
Distance = Speed × Time
Distance = 86 km/h × 0.92 h = 79.12 km
Now, let's calculate the horizontal displacement from the second leg of the journey:
Time = 40 minutes = 40/60 = 0.67 hours
Speed = 99 km/h
To calculate the horizontal component, we need to find the component of the velocity in the east direction. This can be calculated using trigonometry. Since the road is 60 degrees south of east, the component in the east direction will be:
Component in the east direction = Speed × cos(60°)
Component in the east direction = 99 km/h × cos(60°) = 99 km/h × 0.5 = 49.5 km/h
Distance = Component in the east direction × Time
Distance = 49.5 km/h × 0.67 h = 33.165 km
Now that we have the horizontal displacements from both legs of the journey, we can find the total horizontal displacement:
Total Horizontal Displacement = Sum of individual horizontal displacements
Total Horizontal Displacement = 79.12 km + 33.165 km = 112.285 km (rounded to three decimal places)
Next, let's calculate the vertical displacement:
Since we went due east during the first leg of the journey and the second leg is 60 degrees south of east, there is no vertical displacement. Thus, the vertical displacement is 0 km.
Finally, we can calculate the total displacement by combining the horizontal and vertical components:
Total Displacement = √((Total Horizontal Displacement)² + (Vertical Displacement)²)
Total Displacement = √((112.285 km)² + (0 km)²)
Total Displacement = √(12,591.76 km² + 0 km²) = √12,591.76 km² ≈ 112.25 km (rounded to two decimal places)
Therefore, the total displacement is approximately 112.25 km.