An off-roader explores the open desert in her Hummer. First she drives 25 degrees west of north with a speed of 7.0 km/h for 15 minutes, then due east with a speed of 11 km/h for 7.5 minutes. She completes the final leg of her trip in 22 minutes.What is the direction of travel on the final leg? (Assume her speed is constant on each leg, and that she returns to her starting point at the end of the final leg.)
To determine the direction of the final leg, we need to analyze the individual legs of the trip and then add up their respective vectors.
First, let's break down the information given for each leg:
1. Leg 1: The off-roader drives 25 degrees west of north at a speed of 7.0 km/h for 15 minutes.
2. Leg 2: The off-roader drives due east at a speed of 11 km/h for 7.5 minutes.
3. Leg 3: The off-roader completes the final leg in 22 minutes.
To find the direction of the final leg, we'll first calculate the displacement vectors for each leg:
For Leg 1:
- The magnitude of the displacement can be calculated by multiplying the speed (7.0 km/h) by the time (15 minutes), and then converting the time from minutes to hours:
- Displacement magnitude = (7.0 km/h) * (15 minutes / 60 minutes/h) = 1.75 km.
- The direction of the displacement is 25 degrees west of north.
For Leg 2:
- The magnitude of the displacement can be calculated by multiplying the speed (11 km/h) by the time (7.5 minutes), and then converting the time from minutes to hours:
- Displacement magnitude = (11 km/h) * (7.5 minutes / 60 minutes/h) = 1.375 km.
- The direction of the displacement is due east.
For Leg 3:
- The displacement vector is not given explicitly, so we have to calculate it using the given information.
To find the direction of the final leg, we need to add up the displacement vectors for each leg and find the resultant displacement:
We'll denote the final leg's displacement vector as R.
Let's convert the magnitude of the displacements into their x and y components:
For Leg 1:
- x-component = (1.75 km) * cos(25°) = 1.586 km
- y-component = (1.75 km) * sin(25°) = 0.747 km
For Leg 2:
- x-component = (1.375 km) * cos(0°) = 1.375 km
- y-component = (1.375 km) * sin(0°) = 0 km
For Leg 3:
- Let's denote the x-component of the final leg as Rx and the y-component as Ry.
Next, we'll add up the x and y components of the displacement vectors for each leg:
- Rx = (1.586 km + 1.375 km) = 2.961 km
- Ry = (0.747 km + 0 km) = 0.747 km
Now we can use the components to find the magnitude and direction of the resultant displacement:
- Resultant displacement magnitude (R) = sqrt(Rx^2 + Ry^2) = sqrt((2.961 km)^2 + (0.747 km)^2) = 3.058 km
To find the direction, we can use trigonometry:
- Resultant displacement direction = arctan(Ry / Rx) = arctan(0.747 km / 2.961 km) = arctan(0.252)
Converting this angle to degrees, we get:
- Resultant displacement direction ≈ 14.34°
Therefore, the direction of travel on the final leg is approximately 14.34 degrees east of north.