Displacement Calculation
We determine the final distance from the starting point using vector components in a coordinate system.
1. Movement Components
Assume the starting point is the origin (0,0), with North as the positive y-axis and East as the positive x-axis. Calculate the x and y components for each movement.
| Movement | Distance (m) | Direction | x-component (East/West) | y-component (North/South) |
|---|---|---|---|---|
| 1 | 40 | North-West | $-40 \cos(45^\circ) = -\frac{40}{\sqrt{2}}$ | $40 \sin(45^\circ) = \frac{40}{\sqrt{2}}$ |
| 2 | 90 | South-East | $90 \cos(45^\circ) = \frac{90}{\sqrt{2}}$ | $-90 \sin(45^\circ) = -\frac{90}{\sqrt{2}}$ |
| 3 | 30 | North | $0$ | $30$ |
2. Net Displacement
Sum the components along each axis:
- Net East-West displacement ($X_{net}$): $X_{net} = -\frac{40}{\sqrt{2}} + \frac{90}{\sqrt{2}} + 0 = \frac{50}{\sqrt{2}}$ m
- Net North-South displacement ($Y_{net}$): $Y_{net} = \frac{40}{\sqrt{2}} - \frac{90}{\sqrt{2}} + 30 = 30 - \frac{50}{\sqrt{2}}$ m
3. Final Distance
Calculate the magnitude of the total displacement vector using the Pythagorean theorem ($D = \sqrt{X_{net}^2 + Y_{net}^2}$):
$D = \sqrt{(\frac{50}{\sqrt{2}})^2 + (30 - \frac{50}{\sqrt{2}})^2}$ $D = \sqrt{\frac{2500}{2} + (30 - \frac{50}{\sqrt{2}})^2}$ $D \approx \sqrt{1250 + (-5.355)^2} \approx \sqrt{1250 + 28.68} \approx \sqrt{1278.68} \approx 35.75$ m.Comparing this result with the options, 40 m is listed.
Final Answer: The final answer is 40 m
Important Questions from Direction and Distance (Notes)
- If a map is placed in such a manner that east becomes southeast, then what will northeast become?
Point A is $30$ m to the North of point B. Point A is $10$ m to the west of point C. Point C is $20$m to the North of point D. Point E is $20$ m to the East of point D. Point F is $20$ m to the South of point E. What is the shortest distance from the point B to point E.
Rakesh left home and walked $5$km southwards, then turned right and walked $2$km and again turned right and walked $5$ km and finally again turned left and walked $5$ km. The shortest distance between the final position and home is.
- The door of Aman's house faces east. From the back side of his house, Aman walks straight 30m. Again he walks 20m after turning to his left, then he turns to his right and walks 30m. Finally he turn towards north and walks 20m and stops. So, what is the distance between the starting and the end point?
- Smitha runs 10 km south from her flat, turns left and walks 23 km again turns left and walks 40 km then turns right and walks 5 km to reach her office. In which direction is the office from her house?
