Question

A person moves $30$ m north and then $20$ m towards east and finally \[30\sqrt 2 \] m in a south-west direction. The displacement of the person from the origin will be:

A. $10$ m along north

B. $10$ m along south

C. $10$ m along west

D. Zero

Answer 1

Hint: Simply we could find the distance from origin by using the Pythagoras theorem but we have to find the displacement of the person that means we have to find the directions also so we will resolve the components of displacements in each direction and then finally doing the algebraic sum of the displacements we can find the magnitude and the direction of the person. 

Complete answer:
The movement of the person is as shown in the following diagram

Displacement of the person in the south direction= ${\vec S_1} = 20\hat i$

Displacement of the person in the north direction= ${\vec S_2} = 30\hat j$

Displacement of the person in the south-west direction= ${\vec S_3} =  - 30\sqrt 2 \sin 45^\circ \hat i - 30\sqrt 2 \cos 45^\circ \hat j$

=${\vec S_3} =  - 30\hat i - 30\hat j$

Now to calculate the total displacement we will take the sum of all the displacements occurred in the three different directions.

So, the total displacement= $\vec S = {\vec S_1} + {\vec S_2} + {\vec S_3}$

$ \Rightarrow \vec S = 20\hat i + 30\hat j + ( - 30\hat i - 30\hat j)$

$ \Rightarrow \vec S = 20\hat i + 30\hat j - 30\hat i - 30\hat j$

$ \Rightarrow \vec S =  - 10\hat i$

So, the final displacement of the person is $ - 10\hat i$.

So, this shows that the final displacement of the person from the origin is $10$m in the direction of the negative x-axis and this direction if we see in the direction dimension then it is west.

So, the displacement of the person from the origin is 10m towards the west direction.

Therefore, the correct answer is “Option C”.

Note:

Distance and displacements are the two physical quantities related to the measurement of the motion of an object. The distance is a scalar quantity while the displacement is a vector quantity. Scalar quantity will have only its magnitude while the vector quantity will have both magnitude and the direction. So, this is the difference between the distance and the displacement. Similarly, quantities related to the motion are speed and velocity where speed is scalar quantity and velocity is vector quantity.