Question
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
(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
Answers
GPT 3.5
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].
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].