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)
- N sits to the North of L and West of M. L sits to the East of O. If everyone is facing North, which direction of O, is N?
I starts from Point A and drives 5 km towards the north. He then takes a right turn, drives 2 km, turns right and drives 9 km. He then takes a right turn and drives 7 km. He takes a final right turn, drives 4 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are $90^\circ$ turns only unless specified.)
A tourist drives $20 \text{ km}$ towards east, turns right and drives $6 \text{ km}$, then drives $6 \text{ km}$ towards west. He then turns to his left and drives $4 \text{ km}$ and finally turns right and drives $14 \text{ km}$. Where is he from his starting point?
- A Couple started to their offices in the morning from their home. While husband drives 3 km South and then 2 km to his right to reach his office, wife goes 2 km towards East for her office. With respect to husband's office, how far and in which direction is the wife's office ?
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.
