Question

In going from one city to another, a car travels $75km$ north, $60km$ north-west and $20km$ east. The magnitude of displacement between the two cities is (take$\sqrt{2}=0.7$)
$\text{A}\text{. }170km$
$\text{B}\text{. 137km}$
$\text{C}\text{. }119km$
$\text{D}\text{. }140km$

Answer 1

Hint: For finding the displacement of a car, we have to take the distance travelled in directions in the vector form. Take x-axis as the West-East line and y-axis as the North-South line. After finding the displacement of each turn of the car, we will find the magnitude of displacement between two cities.

Formula used:
$s={{s}_{f}}-{{s}_{i}}$

Complete step by step answer:
Displacement is a vector quantity which refers to “how far from a point an object is”, it is an object’s overall change in position. We can say, a displacement is a vector whose length is the shortest distance between the two points. Displacement is calculated by finding the final distance of the object from a fixed point and then subtracting its initial distance from the same point.
Displacement is represented by the symbol $s$
$s={{s}_{f}}-{{s}_{i}}$
$\begin{aligned} & s=\text{ displacement} \\\ & {{s}_{i}}=\text{ initial position} \\\ & {{s}_{f}}=\text{ final position} \\\ \end{aligned}$

Displacement of car in North direction = $75\widehat{j}$
Displacement of car in North-West direction = $-60\cos {{45}^{\circ }}\widehat{i}+60\sin {{45}^{\circ }}\widehat{j}$
Displacement of car in East direction = $20\widehat{i}$
Final position of the car = $\left( \dfrac{-60}{\sqrt{2}}+20 \right)\widehat{i}+\left( \dfrac{60}{\sqrt{2}}+75 \right)\widehat{j}$
$=-22\widehat{i}+117\widehat{j}$
Finding magnitude of the displacement of the car,
$=\sqrt{{{(117)}^{2}}+{{(22)}^{2}}}=\sqrt{14173}=119$
Displacement of the car is $119km$
Hence, the correct option is C.

Additional information:
Distance is the measure of how much ground a body has covered. It is a scalar quantity. Displacement refers to how far a body lies from a fixed point; it is the body's overall change in the position.
Speed is defined as the rate at which a body covers the distance. It is a scalar quantity. Velocity refers to the rate at which a body changes its position. It is a vector quantity.

Note: Students should be careful while calculating the displacement of the object in different directions. Keep in mind that positive x-axis is the East direction while negative x-axis is the West direction. Similarly, when the body is moving towards the North, it is taken along positive y-axis while going towards South is taken along negative y-axis.