A person climbs a hill in a straight path from point 'O' on the ground in the direction of north-east and reaches a point 'A' after travelling a distance of 5 km. Then, from the point 'A' he moves to point 'B' in the direction of north-west. Let the distance AB be 12 km. Now, how far is the person away from the starting point 'O'?
UPSC - 2016
This problem involves vector addition and geometry. We need to determine the final displacement from the starting point. The key is to recognize the directions (north-east, north-west) and the angles they form with the cardinal directions. We can use coordinate geometry or vector methods, but a geometric approach using the Pythagorean theorem will be most straightforward if we can identify a right-angled triangle.
A person walks 12 km due north, then 15 km due east, after that 19 km due west and then 15 km due south. How far is the from the starting point?
UPSC - 2016
This problem involves calculating the displacement from a starting point after a series of movements in different directions. We can break down the movements into components along the North-South axis and the East-West axis. Then, we find the net displacement in each direction and use the Pythagorean theorem to find the straight-line distance from the starting point.
The net movements form a right-angled triangle where the two perpendicular sides are the net North-South displacement and the net East-West displacement. The distance from the starting point is the hypotenuse of this triangle.
Let \(D_N\) be the net displacement North-South and \(D_E\) be the net displacement East-West.
Using the Pythagorean theorem, the distance \(D\) from the starting point is:
\[ D = \sqrt{D_N^2 + D_E^2} \] \[ D = \sqrt{(3 \text{ km})^2 + (4 \text{ km})^2} \] \[ D = \sqrt{9 \text{ km}^2 + 16 \text{ km}^2} \] \[ D = \sqrt{25 \text{ km}^2} \] \[ D = 5 \text{ km} \]So, the person is 5 km from the starting point.
A) 5 km is the correct distance calculated using the net displacements in the North-South and East-West directions and applying the Pythagorean theorem.
Kamal walked 20 m towards north. Then he turned right and walks 30 m. Then he turns right and walks 35 m. Then he turns left and walks 15 m. Finally he turns left and walks 15 m. In which direction and how many metres is he from the starting position?
To solve this problem, we need to track Kamal's movements in the North-South and East-West directions separately. We'll consider North and East as positive directions, and South and West as negative directions. By summing up the displacements in each axis, we can find the final position relative to the starting point.
Correct Option: D) 45 m East
Sakthi was standing facing a pole at the early morning after sunrise, When the shadow of the pole fell to his right means which direction was she facing ?
This question tests basic knowledge of directions and shadows. The key information is the time of day ("early morning after sunrise") and the position of the shadow relative to Sakthi.
Therefore, Sakthi was facing South.
Correct Option: D) South
Kamalesh starts from point A and walks 2 km North, then turns East and walks 3 km. He again turns North and walks 4 km, and finally turns East and walks 5 km to reach point C.
Karunesh starts from a different point B, walks 2 km North, then turns West and walks 3 km. He finally turns North and walks 4 km to reach the same point C where Kamalesh ends.
What is the distance between their starting points A and B?
To find the distance between the starting points A and B, we need to determine the final coordinates of point C relative to A and B. Then, we can find the coordinates of A and B relative to a common origin (or relative to each other) and use the distance formula.
D) 11 Km is the correct distance between the starting points A and B.
Kala cycles 5 km North, then turns East and cycles 4 km, then turns South and cycles 5 km, then turns to her right and cycles 6 km. Where is she now with reference to her starting position
To solve this problem, we need to track Kala's movements in each direction (North, South, East, West) and then calculate her net displacement from the starting point. We can visualize this as a series of vectors or simply keep track of the total distance covered in each cardinal direction.
Therefore, Kala is 2 km West of her starting position.
Correct Option: B) 2 Km West
