Ashish cycles 11 km south, then turns west and cycles 8 km, then turns north and cycles 6 km, then turns east and cycles 2 km, then turns to his left and cycles 5 km. Where is he now with reference to his starting position?
6 km west
Understanding the Cycling Route and Directions
This problem asks us to find the final position of Ashish relative to his starting point after a series of movements in different directions. To solve this, we need to track his displacement in the North-South direction and the East-West direction separately.
Step-by-Step Analysis of Ashish's Movements
Let's break down Ashish's journey step by step:
- Starts at a point (let's call it the origin).
- Cycles 11 km south.
- Turns west and cycles 8 km.
- Turns north and cycles 6 km.
- Turns east and cycles 2 km.
- Turns to his left (from East, left is North) and cycles 5 km.
Calculating Net Displacement
We can analyze the movements along the cardinal directions:
- Northward movements: 6 km (step 4) + 5 km (step 6) = 11 km North.
- Southward movements: 11 km (step 2) = 11 km South.
- Eastward movements: 2 km (step 5) = 2 km East.
- Westward movements: 8 km (step 3) = 8 km West.
Now, let's calculate the net displacement in each axis:
- Net North-South Displacement: Northward displacement - Southward displacement.
$11 \text{ km (North)} - 11 \text{ km (South)} = 0 \text{ km}$
This means his final North-South position is the same as his starting North-South position. - Net East-West Displacement: Eastward displacement - Westward displacement.
$2 \text{ km (East)} - 8 \text{ km (West)} = -6 \text{ km}$
A negative result in this calculation (East - West) indicates a net movement in the West direction. The magnitude is 6 km.
Alternatively, we can consider East as positive and West as negative on one axis, and North as positive and South as negative on the other.
- North/South: $-11 \text{ km (South)} + 6 \text{ km (North)} + 5 \text{ km (North)} = -11 + 11 = 0 \text{ km}$
- East/West: $-8 \text{ km (West)} + 2 \text{ km (East)} = -8 + 2 = -6 \text{ km}$
The net displacement is 0 km in the North-South direction and -6 km in the East-West direction. This means he is 6 km to the West of his starting point.
Summary of Movements
| Movement | Distance (km) | Direction | Contribution to N (+)/S (-) | Contribution to E (+)/W (-) |
|---|---|---|---|---|
| 1 | 11 | South | -11 | 0 |
| 2 | 8 | West | 0 | -8 |
| 3 | 6 | North | +6 | 0 |
| 4 | 2 | East | 0 | +2 |
| 5 | 5 | North (Left from East) | +5 | 0 |
| Total | Net N/S | Net E/W | ||
|---|---|---|---|---|
| $-11 + 6 + 5 = 0$ | $-8 + 2 = -6$ |
Final Position Relative to Starting Point
The net displacement calculations show:
- Net North-South displacement = 0 km
- Net East-West displacement = -6 km (which is 6 km West)
Therefore, Ashish is 6 km west of his starting position.
Revision Table: Direction and Distance Problems
| Concept | Explanation | Key Tip |
|---|---|---|
| Displacement | The shortest distance from the starting point to the ending point, including direction. | Not the total distance traveled. |
| Resolving Movement | Breaking down each movement into North-South and East-West components. | Use a coordinate system (e.g., N=+, S=-, E=+, W=-). |
| Net Displacement | Summing up all movements in the North-South axis and separately in the East-West axis. | Final position is determined by the net change in both axes. |
| Turning Left/Right | Visualize or draw the path. From North, Right is East, Left is West. From East, Right is South, Left is North. From South, Right is West, Left is East. From West, Right is North, Left is South. | Crucial for determining the direction of the next step. |
Additional Information: Solving Direction and Distance Questions
Direction and distance problems are common in aptitude tests. They test your ability to visualize movement and calculate displacement. Here are some tips for solving them effectively:
- Always start by defining a starting point.
- Draw a diagram if necessary. This helps visualize the turns and movements.
- Keep track of movements in the perpendicular directions (North-South and East-West) separately.
- Understand the difference between total distance traveled and displacement. Displacement is the straight-line distance and direction from start to end.
- Pay close attention to the direction of turns (left or right) from the current facing direction.
- For complex paths, summing up displacements along the axes is usually the most reliable method.
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?
