A car leaves point A and drives at 100.0 km/h [E] for 1.2 h. It then heads north at 75.0 km/h for 0.6 h and finally [N42.0°W] at 95.0 km/h for 1.0 h, arriving at point B.
(a) Determine the distance between points A and B.
(b) Determine the displacement from point A to point B.
(c) Determine the average speed between points A and B.
(d) Determine the average velocity from point A to point B
To solve this problem, we'll break the car's journey into three separate legs and calculate the distance, displacement, and average speed for each leg.
First leg (east): The car drives at 100.0 km/h for 1.2 hours. The distance covered is given by speed × time = 100.0 km/h × 1.2 h = 120.0 km.
Second leg (north): The car drives at 75.0 km/h for 0.6 hours. The distance covered is 75.0 km/h × 0.6 h = 45.0 km.
Third leg (northwest): The car drives at 95.0 km/h for 1.0 hour, heading N42.0°W. The distance covered is cos(42°) × 95.0 km/h × 1.0 h = 61.5096 km.
(a) The total distance between points A and B is the sum of the distances covered in each leg:
Total distance = 120.0 km + 45.0 km + 61.5096 km = 226.5096 km.
(b) The displacement is the straight-line distance from point A to point B. To find this, we can calculate the east-west displacement and the north-south displacement separately.
East-west displacement = 120.0 km (positive because it's east)
North-south displacement = cos(42°) × 45.0 km - sin(42°) × 61.5096 km = -3.4523 km
Displacement = √(East-west displacement)² + (North-south displacement)² = √(120.0 km)² + (-3.4523 km)² = √14711.2526 km² ≈ 121.24 km
The displacement from point A to point B is approximately 121.24 km.
(c) The average speed is given by total distance ÷ total time. The total time is the sum of the times for each leg:
Total time = 1.2 h + 0.6 h + 1.0 h = 2.8 h.
Average speed = 226.5096 km ÷ 2.8 h ≈ 80.89 km/h
The average speed between points A and B is approximately 80.89 km/h.
(d) The average velocity is given by the displacement ÷ total time:
Average velocity = 121.24 km ÷ 2.8 h ≈ 43.30 km/h [N25.5°E]
The average velocity from point A to point B is approximately 43.30 km/h [N25.5°E].
(a) To determine the distance between points A and B, we can calculate the total distance traveled in each direction and then sum them up.
Distance traveled in the east direction:
Distance = Speed × Time
Distance = 100.0 km/h × 1.2 h
Distance = 120.0 km
Distance traveled in the north direction:
Distance = Speed × Time
Distance = 75.0 km/h × 0.6 h
Distance = 45.0 km
Distance traveled in the northwest direction:
Distance = Speed × Time
Distance = 95.0 km/h × 1.0 h
Distance = 95.0 km
Total distance between points A and B:
Total Distance = Distance in east + Distance in north + Distance in northwest
Total Distance = 120.0 km + 45.0 km + 95.0 km
Total Distance = 260.0 km
Therefore, the distance between points A and B is 260.0 km.
(b) To determine the displacement from point A to point B, we need to calculate the straight-line distance between these two points.
Displacement = √((Distance north)^2 + (Distance west)^2)
Displacement = √(45.0 km^2 + 95.0 km^2)
Displacement = √(2025 km^2 + 9025 km^2)
Displacement = √(11050 km^2)
Displacement ≈ 105.13 km
Therefore, the displacement from point A to point B is approximately 105.13 km.
(c) To determine the average speed between points A and B, we need to calculate the total distance traveled and divide it by the total time taken.
Total Distance = 260.0 km
Total Time = 1.2 h + 0.6 h + 1.0 h
Total Time = 2.8 h
Average Speed = Total Distance / Total Time
Average Speed = 260.0 km / 2.8 h
Average Speed ≈ 92.86 km/h
Therefore, the average speed between points A and B is approximately 92.86 km/h.
(d) To determine the average velocity from point A to point B, we need to calculate the displacement and divide it by the total time taken.
Displacement = 105.13 km
Total Time = 2.8 h
Average Velocity = Displacement / Total Time
Average Velocity = 105.13 km / 2.8 h
Average Velocity ≈ 37.55 km/h
Therefore, the average velocity from point A to point B is approximately 37.55 km/h.
To solve this problem, we can break it down into smaller steps:
Step 1: Calculate the distance traveled in each segment.
Step 2: Calculate the displacement from point A to point B.
Step 3: Calculate the average speed between points A and B.
Step 4: Calculate the average velocity from point A to point B.
Let's solve it step by step:
Step 1: Calculate the distance traveled in each segment.
- In the first segment, the car drives at 100.0 km/h [E] for 1.2 hours. Distance = speed × time = 100.0 km/h × 1.2 h = 120.0 km.
- In the second segment, the car drives north at 75.0 km/h for 0.6 hours. Distance = speed × time = 75.0 km/h × 0.6 h = 45.0 km.
- In the third segment, the car drives [N42.0°W] at 95.0 km/h for 1.0 hour. We need to decompose this velocity into its north and west components. The north component is 95.0 km/h × sin(42.0°) = 64.69 km/h, and the west component is 95.0 km/h × cos(42.0°) = 72.18 km/h. Therefore, the distance traveled in this segment is the hypotenuse of the right triangle formed by the north and west components. Distance = √((64.69 km)^2 + (72.18 km)^2) = 96.86 km.
Step 2: Calculate the displacement from point A to point B.
To find the displacement, we need to add up the displacement in each segment. The displacement in the first segment is 120.0 km to the east. In the second segment, the displacement is 45.0 km to the north. In the third segment, the displacement is 96.86 km in the direction [N42.0°W]. To find the total displacement, we can use vector addition. We need to decompose the third segment's displacement into its north and west components: north = 96.86 km × sin(42.0°) = 64.69 km, and west = 96.86 km × cos(42.0°) = 72.18 km. The total displacement is then the sum of the east, north, and west components. Displacement = 120.0 km [E] + 45.0 km [N] + 72.18 km [W] = 93.18 km [35.7°N of W].
Step 3: Calculate the average speed between points A and B.
Average speed is defined as the total distance traveled divided by the total time taken. The total distance traveled is the sum of the distances calculated in step 1: 120.0 km + 45.0 km + 96.86 km = 261.86 km. The total time taken is the sum of the times in each segment: 1.2 hours + 0.6 hours + 1.0 hour = 2.8 hours. Average speed = total distance ÷ total time = 261.86 km ÷ 2.8 h = 93.52 km/h.
Step 4: Calculate the average velocity from point A to point B.
Average velocity is defined as the displacement divided by the total time taken. The displacement from step 2 is 93.18 km [35.7°N of W]. We already calculated the total time taken in step 3: 2.8 hours. Average velocity = displacement ÷ total time = 93.18 km ÷ 2.8 h = 33.28 km/h [35.7°N of W].
Summary of results:
(a) The distance between points A and B is 261.86 km.
(b) The displacement from point A to point B is 93.18 km [35.7°N of W].
(c) The average speed between points A and B is 93.52 km/h.
(d) The average velocity from point A to point B is 33.28 km/h [35.7°N of W].