Rahul starts walking towards the east and covers a distance of 80 m. He turns to his right and walks 30 m. Again, he turns right and walks 30 m. Finally, he turns towards the north and covers a distance of 30 m. How far is he now from his original position?
50 m
Understanding Distance and Direction Problems
Distance and direction problems test your ability to follow a series of movements and determine the final position relative to the starting point. It's often helpful to visualize the path or draw a simple diagram.
Step-by-Step Movement Analysis
Let's track Rahul's journey step by step from his original position.
- Starts at the origin: Assume Rahul starts at point O (Original Position).
- Moves East 80 m: Rahul walks 80 m towards the east. Let's say he reaches point A. His position is now 80 m east of O.
- Turns Right, Walks 30 m: From point A (East), turning right means turning towards the south. He walks 30 m south. Let's say he reaches point B. His position is now 80 m east and 30 m south of O.
- Turns Right, Walks 30 m: From point B (facing South), turning right means turning towards the west. He walks 30 m west. Let's say he reaches point C. He moved 30 m west from A, which was 80 m east of O. So, his new east position relative to O is \(80 - 30 = 50\) m East. He is still 30 m south of O. His position is now 50 m east and 30 m south of O.
- Turns North, Covers 30 m: From point C (facing West), turning north means moving upwards in our visualization. He walks 30 m north. He was 30 m south of O. Moving 30 m north cancels out the southward movement. Let's say he reaches point D (Final Position). His position is now 50 m east and \(30 - 30 = 0\) m south/north of O.
Calculating the Final Position
Let's represent the directions using coordinates, with the starting point at (0,0). East is positive x, North is positive y.
- Starting Point: (0, 0)
- Move East 80 m: (0 + 80, 0) = (80, 0)
- Turn Right (South), Walk 30 m: (80, 0 - 30) = (80, -30)
- Turn Right (West), Walk 30 m: (80 - 30, -30) = (50, -30)
- Turn North, Cover 30 m: (50, -30 + 30) = (50, 0)
Rahul's final position is at coordinates (50, 0).
Calculating Distance from Original Position
The original position was (0, 0). The final position is (50, 0).
To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
Here, \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (50, 0)\).
Distance = \(\sqrt{(50 - 0)^2 + (0 - 0)^2}\)
Distance = \(\sqrt{(50)^2 + (0)^2}\)
Distance = \(\sqrt{2500 + 0}\)
Distance = \(\sqrt{2500}\)
Distance = \(50\)
So, Rahul is 50 meters away from his original position.
Summary of Movements
The movements can be summarized in a table:
| Movement | Direction | Distance | Change in Position (x, y) | Current Position (relative to start) |
|---|---|---|---|---|
| Start | - | - | (0, 0) | (0, 0) |
| 1 | East | 80 m | (+80, 0) | (80, 0) |
| 2 (Right turn) | South | 30 m | (0, -30) | (80, -30) |
| 3 (Right turn) | West | 30 m | (-30, 0) | (50, -30) |
| 4 (North turn) | North | 30 m | (0, +30) | (50, 0) |
The final position is (50, 0) which is 50 m east of the original position.
Revision Table: Key Concepts
| Concept | Explanation | Application in Problem |
|---|---|---|
| Cardinal Directions | North, South, East, West. Used to describe movement path. | Movements are described using East, Right turn (South), Right turn (West), North turn. |
| Turns (Right/Left) | A turn changes the direction of movement based on the current facing direction. Right turn is 90° clockwise, Left turn is 90° anti-clockwise. | Right from East is South. Right from South is West. Turning towards North explicitly changes direction. |
| Displacement vs. Distance | Displacement is the straight-line distance and direction from start to end. Distance is the total path length covered. This problem asks for displacement magnitude. | We calculated the straight-line distance from (0,0) to (50,0). |
| Coordinate System | Using (x, y) coordinates simplifies tracking position relative to an origin. | Start at (0,0). East is +x, West is -x, North is +y, South is -y. |
Additional Information: Direction Concepts
Understanding how turns affect direction is crucial in distance and direction problems.
- If you are facing North: Right is East, Left is West.
- If you are facing East: Right is South, Left is North.
- If you are facing South: Right is West, Left is East.
- If you are facing West: Right is North, Left is South.
In this problem:
- Start (East): Turn Right → South
- Facing South: Turn Right → West
- Finally turns towards North: This means the next movement is in the North direction regardless of the previous facing direction (West).
Visualizing or sketching the path helps avoid confusion with turns.
Similar Questions
After leaving school, Hemant and Anurag walk towards the north for 1 km to reach a bus stop. From the bus stop, Hemant turns left and walks for 500 m. Then he turns towards his left and walks for 2 km to reach his home. From the bus stop, Anurag turns right and walks for 250 m. Then he turns right and walks for 1 km to reach his home. In which direction is Anurag's home from Hemant's home?
Tamanna is walking towards the north from her house. After 500 metres she turns left and walks for 500 metres. Then she turns left and walks 250 metres to reach her friend’s house. In Which direction is her house from her friend’s house?
Starting from a point A, an animal walked 15 m towards East. It took a left turn and walked 25 m. Then, it took a right turn and walked 10 m. Again it took a left turn and walked 15 m. Finally, it took a left turn and walked 25 m. How far and in which directions is the animal now from the starting point A?
- Sumit is facing north-east, and then he turns 90° clockwise. After that, he turns 135° anticlockwise. Finally, he turns 90° clockwise. In which direction is he facing now?
The length and breadth of a hall are 16 m and 13 m, respectively. A dog runs along all four walls once to catch the bone. How much total distance does the dog cover?
Gurman drove 28 km towards the West, took a left turn and drove 22 km. She then took a right turn and drove 18 km. She again took a right turn and drove 22 km. How far is she from starting point in a straight line?
Indira starts from point A, which is 7 km towards East of point B. From point A, she drives 4 km towards East. From here, she takes a left turn and drives 9 km. She then takes a left turn, drives 9 km and stops at point C. If from point C, Indira drives 9 km towards South, how far will she be from point B?
Shalini drives 5 km North, then she turns East and drives 12 km. After this, she turns North and drives 4 km. Again she turns East and drives 5 km. What is the total distance covered by Shalini?
Starting from point X, a person walked 20 m towards South to reach point Y. From there, he took a left turn and walked 25 m to reach point Z. He then took a left turn and walked for another 20 m to reach point P. What is the approximate shortest distance between the points X and Z and in which direction is the person's final position with respect to point X?
Stepping outside her school, Katrina walked 40 m towards South to reach an ice - cream parlor. From there, she took a left turn and walked 15 m to reach a bus stop. She then took a left - turn and walked for another 30 m to reach the central park. What is the approximate shortest distance between the ice - cream parlor and the central park?
Important Questions from Direction and Distance Turns
After leaving school, Hemant and Anurag walk towards the north for 1 km to reach a bus stop. From the bus stop, Hemant turns left and walks for 500 m. Then he turns towards his left and walks for 2 km to reach his home. From the bus stop, Anurag turns right and walks for 250 m. Then he turns right and walks for 1 km to reach his home. In which direction is Anurag's home from Hemant's home?
A bank employee drives 10 km towards South from her house and turns to her left and drives another 20 km. She again turns left and drives 40 km, then she turns to her right and drives for another 5 km. She again turns to her right and drives another 30 km to reach her bank where she works. What is the shortest distance between her bank and her house ?
A woman runs 12 km towards her North, then 6 km towards her South and then 8 km towards her East. In which direction is she from her starting point ?
Two friends Xand Ystart running and they run together for 50 m in the same direction and reach a point. Xturns right and runs 60 m, while Yturns left and runs 40 m. Then Xturns left and runs 50 m and stops, while Yturns right and runs 50 m and then stops. How far are the two friends from each other now?
Shekhar starts walking south from his school and then took one right turn, one left turn and one right turn to reach his shop. In which direction is Shekhar with respect to his school?
