A car travels east at 82 km/h for 1.5 h. It then travels 30.0° east of north at 113 km/h for 1.6 h.
What is the average velocity for the trip? (Magnitude and Direction)
You have to find the vector displacement (by finding the position vectors then adding them)
then
avgvelocity=displacement/time
To find the average velocity for the trip, we need to determine both the magnitude and direction of the average velocity.
To calculate the magnitude of the average velocity, we can use the formula:
Average velocity = Total displacement / Total time
First, let's calculate the displacement for each segment of the trip:
For the first segment, the car travels east at a constant speed of 82 km/h for 1.5 hours. The displacement in the east direction can be calculated as:
Displacement1 = Speed1 × Time1 = 82 km/h × 1.5 h = 123 km east
For the second segment, the car travels at a speed of 113 km/h at an angle of 30° east of north for 1.6 hours. To find the displacement, we first need to calculate the horizontal and vertical components of the displacement:
Horizontal displacement = Speed2 × cos(theta) × Time2
Vertical displacement = Speed2 × sin(theta) × Time2
where theta is the angle of 30° east of north.
Horizontal displacement = 113 km/h × cos(30°) × 1.6 h = 97.95 km
Vertical displacement = 113 km/h × sin(30°) × 1.6 h = 54.92 km
Now, let's calculate the resultant displacement by calculating the sum of the eastward displacements and the sum of the northward displacements:
Sum of eastward displacements = Displacement1 + Horizontal displacement = 123 km + 97.95 km = 220.95 km east
Sum of northward displacements = Vertical displacement = 54.92 km north
To find the total displacement, we can use the Pythagorean theorem:
Total displacement = √[(Sum of eastward displacements)^2 + (Sum of northward displacements)^2]
= √[(220.95 km)^2 + (54.92 km)^2]
= √(48,813.2025 km^2 + 3,018.4864 km^2)
= √51,831.689 km^2
= 227.69 km
Next, we need to calculate the total time taken for the entire trip:
Total time = Time1 + Time2 = 1.5 h + 1.6 h = 3.1 h
Finally, we can calculate the magnitude of the average velocity:
Average velocity = Total displacement / Total time
= 227.69 km / 3.1 h
≈ 73.48 km/h
So, the magnitude of the average velocity for the trip is approximately 73.48 km/h.
To find the direction of the average velocity, we can use trigonometry. The direction can be determined by calculating the angle between the resultant displacement vector and the east direction.
Angle = arctan((Sum of northward displacements) / (Sum of eastward displacements))
= arctan(54.92 km / 220.95 km)
= arctan(0.249)
≈ 14.16°
Therefore, the direction of the average velocity is approximately 14.16° north of east.
In summary, the average velocity for the trip is approximately 73.48 km/h directed 14.16° north of east.