Question

An airplane is flying with a velocity of $50\sqrt 2 km/h$ in the north-east direction. Wind is blowing at $25\;km/h$ from north to south. What is the resultant displacement?

Answer 1

Hint:To find the resultant of the given velocity vectors, we can draw a phase diagram of the two velocity vectors as shown below. Since velocity is a vector component, to add two or more vectors we can use the triangle addition of vectors as shown below.

Formula used:
$\vec{OA}+\vec{AB}=\vec{OB}$, use the diagram below for reference


Complete step-by-step solution:
We know that vectors are quantities, which have both direction and magnitude, thus to add two or more vectors, we use the parallelogram law of vector addition. Or we can also use the triangle law of vector addition, which is a simplified version of the parallelogram law of vector addition .
The triangle law of vector addition states the following:
“when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector”
Consider the diagram as shown below:

Where, $\vec {OA}=50\sqrt 2km/h$ gives the velocity vector of the airplane , and $\vec{AB}=25km/h$ is the velocity vector of the wind and $\vec {OB}$ is the resultant vector which we need to find.
Then the magnitude of the resultant is given as
$OB=\sqrt{{OA}^2+{AB}^2}$
$\implies OB=\sqrt{(50\sqrt 2)^2+(25)^2}$
$\implies OB=\sqrt { 5000+625}$
$\implies OB=\sqrt{5625}$
$\therefore OB= 75 km/h$
Thus, $\vec{OB}= 75 km/h$ along the west to east direction is the correct answer.

Note: Using parallelogram law of vector addition, we can also subtract two or more vectors. To account for the negative sign, we can invert the direction of the vector with the negative sign and then add them together. Also note that the vectors can be displaced keeping the direction and the magnitude the same.