Salman starts from his office and drives 29 km towards South. Then he turns right and drives 16 km, again he turns right and drives 29 km. Finally he turns right and drives 13 km. At what distance is he from his start point in a straight line?
3 km
Calculating Distance from Start Point: A Step-by-Step Approach
This question asks us to find the straight-line distance of Salman from his starting point (his office) after completing a series of movements involving different directions and turns. To solve this, we need to trace his path and determine his final position relative to the start.
Understanding Directions and Turns
Let's first understand how the turns affect the direction of movement:
- Starting point: Office
- First move: 29 km towards South.
- First turn: Turns right. If facing South, a right turn means facing West.
- Second move: 16 km towards West.
- Second turn: Turns right. If facing West, a right turn means facing North.
- Third move: 29 km towards North.
- Third turn: Turns right. If facing North, a right turn means facing East.
- Fourth move: 13 km towards East.
Tracing Salman's Journey and Displacement
We can visualize Salman's path and see how his position changes relative to the starting point. Let's consider movements along the North-South axis and the East-West axis separately.
- Initial position: Office (Let's consider this as the origin, position (0,0)).
- Move 1: 29 km South. His position is now 29 km South of the office.
- Move 2: Turns right and drives 16 km. This is 16 km West from his current position (29 km South of office). So, he is now 29 km South and 16 km West of the office.
- Move 3: Turns right and drives 29 km. This is 29 km North from his current position (which was 29 km South and 16 km West of office). The 29 km North movement cancels out the initial 29 km South movement. So, his position is now effectively 0 km North/South relative to the original South movement, and still 16 km West of the office's East-West line. His position is 16 km West of the office.
- Move 4: Finally turns right and drives 13 km. This is 13 km East from his current position (16 km West of office). His current position was 16 km West of the office. Moving 13 km East from there means he is now \(16 - 13\) km West of the office.
Calculating the Net Displacement
Let's look at the net effect of the movements in each direction:
- North-South Movement: He moved 29 km South and then 29 km North. The net South-North displacement is \(29 \text{ km South} - 29 \text{ km North} = 0 \text{ km}\). He is at the same North-South level as his starting point (the office).
- East-West Movement: After the first right turn (facing South), he moved 16 km West. After the third right turn (facing North), he moved 13 km East. The net East-West displacement is \(16 \text{ km West} - 13 \text{ km East} = 3 \text{ km West}\). He is 3 km West of his starting point.
Final Distance from the Start Point
Salman's final position is 0 km in the North-South direction and 3 km in the West direction relative to his office. Therefore, he is 3 km West of his office.
The straight-line distance from the start point is simply the magnitude of this net displacement.
Distance from start point = \(3 \text{ km West}\)
The straight-line distance is 3 km.
| Move | Direction | Distance | Relative to Office (E-W) | Relative to Office (N-S) |
|---|---|---|---|---|
| Start | - | 0 km | 0 km | 0 km |
| 1 | South | 29 km | 0 km | 29 km South |
| 2 (Right) | West | 16 km | 16 km West | 29 km South |
| 3 (Right) | North | 29 km | 16 km West | (29 South - 29 North) = 0 km |
| 4 (Right) | East | 13 km | (16 West - 13 East) = 3 km West | 0 km |
His final position is 3 km West of his starting point. The straight-line distance is 3 km.
Revision Table: Key Concepts
| Concept | Explanation | In this Problem |
|---|---|---|
| Direction | Cardinal points like North, South, East, West. | South, Right turn (West), Right turn (North), Right turn (East). |
| Right Turn | Turning 90 degrees clockwise from the current direction. | South → West, West → North, North → East. |
| Displacement | The straight-line distance and direction from the start point to the end point. | Net movement in East-West and North-South directions. |
| Distance | The total length of the path traveled. (Not asked in this question, but different from displacement magnitude). | 29 + 16 + 29 + 13 = 87 km total distance traveled. |
Additional Information: Distance vs. Displacement
It is important to distinguish between distance and displacement in physics and related problems.
- Distance: This is a scalar quantity that refers to the total length of the path covered during motion. It only has magnitude. For example, Salman traveled a total distance of \(29 + 16 + 29 + 13 = 87\) km.
- Displacement: This is a vector quantity that refers to the change in position from the starting point to the ending point. It has both magnitude and direction. In this problem, Salman's displacement is 3 km West from his office. The magnitude of the displacement is the straight-line distance from the start point, which is 3 km.
Understanding the difference between distance and displacement is crucial for solving many problems involving motion and directions.
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