Sunita rode her scooty Northward, then turned left and then again rode to her left 4 km. She found herself exactly 2 km. West to her starting point. How far did she ride Northwards initially?
4 km
Understanding the Directional Movement Problem
This problem involves analyzing movement in different directions and determining an unknown distance based on the final position relative to the starting point. Sunita's journey involves moving North, turning Left (which means West from a Northward path), and turning Left again (which means South from a Westward path).
Step-by-Step Analysis of Sunita's Scooty Ride
Let's break down Sunita's movement:
- Sunita starts at a point (let's call it the Origin).
- She rides Northward initially. Let the distance she rode North be \(x\) km. Her position is now \(x\) km North of the Origin.
- She turns Left. Since she was moving North, turning Left means she is now moving West. She rides some distance West (the problem doesn't explicitly state this distance, let's call it \(y\) km). Her position is now \(y\) km West and \(x\) km North of the Origin.
- She turns Left again. Since she was moving West, turning Left means she is now moving South. She rides 4 km South.
Calculating Displacement and Final Position
We can represent the movement using coordinates, assuming the starting point is (0,0).
- Starting Point: (0, 0)
- Move North \(x\) km: Position becomes (0, \(x\))
- Turn Left (West) and move \(y\) km: Position becomes (-\(y\), \(x\))
- Turn Left again (South) and move 4 km: From (-\(y\), \(x\)), moving South 4 km means decreasing the y-coordinate by 4. Position becomes (-\(y\), \(x\) - 4).
The problem states that her final position is exactly 2 km West of her starting point. The starting point is (0,0). A point 2 km West of (0,0) is (-2, 0) on our coordinate system.
So, the final position (-\(y\), \(x\) - 4) must be equal to (-2, 0).
Solving for the Unknown Distance
By equating the coordinates of the final position, we get two equations:
- Equating the x-coordinates: \(-\(y\) = -2\)
- Equating the y-coordinates: \(x - 4 = 0\)
Let's solve these equations:
- From equation 1: \(-\(y\) = -2\). Multiplying both sides by -1, we get \(y = 2\). This means she rode 2 km West after her first turn.
- From equation 2: \(x - 4 = 0\). Adding 4 to both sides, we get \(x = 4\). This \(x\) is the distance she rode Northwards initially.
Final Answer Determination
The question asks for the distance she rode Northwards initially, which we defined as \(x\). We found that \(x = 4\) km.
Therefore, Sunita rode 4 km Northwards initially.
| Action | Direction/Distance | Change in Position | Current Position (assuming Start = (0,0)) |
|---|---|---|---|
| Start | - | - | (0, 0) |
| Ride North | North, \(x\) km | \(\Delta y = +x\) | (0, \(x\)) |
| Turn Left, Ride West | West, \(y\) km | \(\Delta x = -y\) | (-\(y\), \(x\)) |
| Turn Left, Ride South | South, 4 km | \(\Delta y = -4\) | (-\(y\), \(x\)-4) |
| Final Position | 2 km West of Start | - | (-2, 0) |
Equating the final calculated position (-\(y\), \(x\)-4) with the given final position (-2, 0) confirms our values for \(x\) and \(y\).
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Cardinal Directions | North, South, East, West. Represented as directions on a map or coordinate plane. |
| Turns | Turning 'Left' or 'Right' changes the direction of movement relative to the current direction.
|
| Displacement | The shortest distance and direction from the starting point to the ending point. It's a vector quantity. |
| Total Distance | The sum of the lengths of all the paths traveled. It's a scalar quantity. This question asks for a specific part of the total distance. |
| Coordinate System | Using x and y axes (often East-West and North-South) to represent positions and movements makes solving these problems easier. |
Additional Information on Direction and Distance Problems
Problems involving directions and distances are common in aptitude tests and physics. They often require visualizing the path taken and sometimes using the Pythagorean theorem or coordinate geometry to find the final displacement or an unknown distance.
In this specific problem, the key was correctly interpreting the "Left" turns and setting up equations based on the final displacement from the start.
- Going North changes the y-coordinate.
- Going West changes the x-coordinate (typically negative).
- Going South changes the y-coordinate (typically negative).
- Going East changes the x-coordinate (typically positive).
By representing the start as (0,0), moving North is (0, +distance), West is (-distance, 0), South is (0, -distance), and East is (+distance, 0). Combined movements lead to adding these displacement vectors.
Important Questions from Direction Sense Test
A man moves to the east side, then turns left, then right, and then further turns right again and then moves left. Now, in which direction is the man?
Pinky walks a distance 600m towards east, turns left moves 500m, then again turns left and walks 600m and then turns left again and moves 500m and halts. At what distance (in meters) is she from the starting point?
One evening before sunset, two friends Shahrukh and Rakesh were talking to each other face-to-face. If Rakesh’s shadow is exactly to his right side, which direction was Shahrukh facing?
A girl is facing north. She turns 90° in the anti-clockwise direction and then 45° in the clockwise direction. Which direction is she facing now?
A boy runs 4 km to East side, then turns right and runs 6 km, and then turns left and runs 8 km. Again turns left to run 3 km, again turns right 4 km, then turns right 1 km. Now the boy is in which direction?
