You drive due east at 78 km/h for 55 min. Then you follow a road going 60 degrees south of east, going 99 km/h for another 38 min .
Find your total displacement.
To find the total displacement, we need to break down the motion into its north-south (vertical) and east-west (horizontal) components.
First, let's analyze the eastward (horizontal) displacement during the first part of the motion, which is driving due east at 78 km/h for 55 minutes. The formula to calculate displacement in the horizontal direction is:
Displacement_horizontal = velocity_horizontal * time
Since the car is driving due east (which is the positive x-axis), the eastward velocity is positive, and we can simply plug in the given values:
Displacement_horizontal = 78 km/h * (55 min / 60 min/h)
Displacement_horizontal = 71.5 km
Next, let's analyze the southeastward (diagonal) displacement during the second part of the motion, which is following a road 60 degrees south of east at 99 km/h for 38 minutes. To find the horizontal and vertical components of this displacement, we need to use trigonometry.
The horizontal component can be found using cosine:
Displacement_horizontal2 = displacement * cos(angle)
Displacement_horizontal2 = 99 km/h * (38 min / 60 min/h) * cos(60 degrees)
Displacement_horizontal2 = 95.77 km
The vertical component can be found using sine:
Displacement_vertical = displacement * sin(angle)
Displacement_vertical = 99 km/h * (38 min / 60 min/h) * sin(60 degrees)
Displacement_vertical = 85.92 km
Now, we can find the net horizontal displacement by adding the two horizontal components:
Net_horizontal_displacement = Displacement_horizontal + Displacement_horizontal2
Net_horizontal_displacement = 71.5 km + 95.77 km
Net_horizontal_displacement = 167.27 km
And, we can find the net vertical displacement by adding the two vertical components:
Net_vertical_displacement = Displacement_vertical
Net_vertical_displacement = 85.92 km
Finally, we can use the Pythagorean theorem to find the total displacement:
Total_displacement = √(Net_horizontal_displacement^2 + Net_vertical_displacement^2)
Total_displacement = √(167.27^2 + 85.92^2)
Total_displacement = √(27979.43 + 7379.14)
Total_displacement = √(35358.57)
Total_displacement = 187.89 km
Therefore, the total displacement is approximately 187.89 km.