How to Calculate Distance and Displacement - WORKED EXAMPLE - GCSE Physics
TLDRThe video script details a hiker's peculiar journey, covering a total distance of eight kilometers through a series of eastward and northward movements, followed by a southward turn. The hiker's final displacement, calculated using trigonometry and Pythagoras' theorem, is determined to be 3.60 kilometers at a bearing of 056 degrees, illustrating the distinction between total distance traveled and the straight-line distance from the starting point. The script employs a sketch to visualize the hiker's path and employs mathematical methods to find the displacement, emphasizing the importance of both distance and direction in understanding movement.
Takeaways
- 🚶 The hiker's walk involved moving 2 km east, 3.5 km north, and 1 km east again, followed by 1.5 km south.
- 📏 The total distance traveled by the hiker was calculated to be 8.0 kilometers.
- 📐 To determine the final displacement, a sketch was used to visualize the hiker's path and form a right-angled triangle.
- 📏 The displacement was found to be approximately 3.5 kilometers by measuring the hypotenuse of the triangle.
- 🧮 Pythagoras' theorem was applied to calculate the exact displacement as 3.60 kilometers.
- 📐 The direction of the displacement was determined using trigonometry, specifically the tangent function.
- 📐 The angle theta was calculated to be 33.7 degrees, which is the angle north of east from the starting point to the final position.
- 🧭 The bearing, or angle alpha, was determined to be 56.3 degrees, which is the direction from the starting point to the final position.
- 🚷 The difference between total distance (8.0 km) and displacement (3.60 km) highlights the distinction between the two concepts.
- 📐 The use of a scale drawing or a vector diagram can simplify the process of finding displacement and direction.
- 📊 A calculator or a computer could be used to find the exact values of displacement and direction, avoiding manual measurements.
- 📚 The script demonstrates the practical application of mathematical concepts like Pythagoras' theorem and trigonometry in real-world scenarios.
Q & A
What was the total distance traveled by the hiker?
-The hiker traveled a total distance of 8 kilometers.
How did the hiker's walk start in terms of direction?
-The hiker started by walking 2 kilometers east.
How far did the hiker walk north before changing direction?
-The hiker walked 3.5 kilometers north.
What was the hiker's next direction after walking north?
-After walking north, the hiker walked another 1 kilometer east.
In which direction did the hiker walk last, and for how long?
-The hiker walked 1.5 kilometers south as the final leg of their walk.
What is the difference between distance and displacement?
-Distance is the total length of the path traveled, regardless of direction, while displacement is the straight-line distance from the starting point to the final position, along with the direction.
How was the hiker's final displacement calculated?
-The final displacement was calculated using the Pythagorean theorem to find the hypotenuse of a right-angled triangle formed by the eastward, northward, and southward distances.
What was the hiker's total displacement from the starting position?
-The hiker's total displacement was 3.6 kilometers.
How was the direction of the hiker's displacement determined?
-The direction of displacement was determined using trigonometry, specifically the tangent function to find the angle (theta), and then calculating the bearing (angle alpha).
What bearing did the hiker end up from their starting position?
-The hiker ended up at a bearing of 056 degrees, which is 56 degrees north of East.
Why might the bearing be more useful than the angle theta for some applications?
-The bearing is more useful because it provides a direction in terms of the cardinal points (North, South, East, West), which is easier to understand and apply in navigation.
How did the hiker's walk resemble a vector diagram?
-The hiker's walk resembled a vector diagram because each leg of the walk represented a vector with a specific direction and magnitude, and the final position was determined by the sum of these vectors.
Outlines
🚶 Calculating Total Distance and Displacement
The paragraph describes a hiker's journey, detailing the distances traveled in different directions: 2 km east, 3.5 km north, and 1 km east again followed by 1.5 km south. The task is to calculate the total distance traveled, which is the sum of all individual distances, and the final displacement, which is the straight-line distance from the starting point to the ending point. The total distance is calculated to be 8 km. To find the displacement, a rough sketch is used to visualize the path, and trigonometry is applied to find the hypotenuse of a right-angled triangle formed by the hiker's path segments, resulting in an approximate displacement of just over 3.5 km.
📐 Using Trigonometry to Determine Displacement Direction
This paragraph focuses on finding the direction of the hiker's displacement using trigonometric functions. By identifying the right-angled triangle formed by the hiker's path, the angle of displacement (theta) is calculated using the tangent function, which gives an angle of approximately 33.7 degrees. To express this in terms of a bearing, the angle alpha is found by subtracting theta from 90 degrees, resulting in a bearing of 56.3 degrees. The final displacement is thus 3.6 kilometers at a bearing of 056 degrees, illustrating the difference between total distance, which is the length of the path taken, and displacement, which is the straight-line distance in a specific direction from the start to the end point.
Mindmap
Keywords
💡Hiker
💡Distance Traveled
💡Displacement
💡Bearing
💡Trigonometry
💡Pythagoras' Theorem
💡Vector Diagram
💡Right-Angled Triangle
💡Hypotenuse
💡Tangent (Trigonometric Function)
💡SOHCAHTOA
Highlights
The hiker has taken a peculiar path, moving 2 km east, 3.5 km north, another km east, and finally 1.5 km south.
The total distance traveled by the hiker is calculated to be 8 kilometers.
A sketch is used to visualize the hiker's start and end positions, employing a scale of 2 cm to 1 km.
The final displacement is estimated to be just over 3.5 kilometers by visual inspection of the sketch.
Pythagoras' theorem is applied to calculate the exact displacement, forming a right-angled triangle with the hiker's path.
The calculated displacement using trigonometry is 3.6 kilometers.
The difference between total distance and displacement is emphasized, highlighting a common misunderstanding.
Trigonometric functions SOHCAHTOA are used to find the direction of displacement.
The angle theta is calculated to be 33.7 degrees, indicating the direction of displacement.
The bearing angle alpha is derived as 56.3 degrees, providing a practical direction from the starting point.
The final displacement is given as 3.6 kilometers at a bearing of 056 degrees.
The transcript demonstrates the use of mathematics and trigonometry to solve real-world navigation problems.
An alternative method using a protractor for more accurate measurements is suggested for complex diagrams.
The importance of understanding both the magnitude and direction of displacement is discussed.
The transcript provides a clear example of how to calculate displacement over distance in navigation.
The use of a scale drawing and trigonometric calculations allows for a more precise determination of displacement.
The practical application of this method could be useful for hikers, navigators, or anyone needing to calculate travel paths.
The transcript concludes by emphasizing the educational value in understanding the difference between distance and displacement.
Transcripts
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