Question

The pilot of an aircraft flies due east relative to the ground in a wind blowing 20.0km/htoward the south. If the speed of the aircraft in the absence of wind is 70.0 km/h, what is the speed of the aircraft relative to the ground?

Short answer

Speed of the aircraft relative to the ground is 67.08km/h .

Step-by-step solution

The given data

The net and y- component of velocity, considering the east direction as x-component,

$$\begin{gathered} V\left(or{V}_{est}\right)=70.0\mathrm{km}/\mathrm{h} \\ {\mathrm{V}}_{\mathrm{y}}\left(\mathrm{or}{\mathrm{V}}_{\mathrm{south}}\right)=20.0\mathrm{km}/\mathrm{h} \end{gathered}$$

Understanding the concept of velocity

The velocity is a vector quantity. The vector has magnitude as well as direction. We can add the vectors using the law of vector addition. The magnitude of the resultant vector is found by adding the square of the components of the resultant and finding the square root.

We are given two perpendicular velocities; hence, we can find the resultant velocity of aircraft by using the resultant velocity equation.

Formula:

The resultant velocity of a body in motion,$V=\sqrt{V_x^2+V_y^2}$ (i)

Here,v is resultant velocity,$V_x$ is x component of the velocity,$V_y$ is y component of the velocity.

Calculation for aircraft relative to the ground

Using equation (i), we get the resultant velocity.

$$\begin{gathered} V=\sqrt{V_x^2+V_y^2} \\ {V}^2=V_x^2+V_y^2 \\ {V}_x^2=V^2-V_y^2 \\ V=\sqrt{V^2-V_y^2} \end{gathered}$$

Substitute the value of resultant velocity and y component of the velocity in the above equation.

$$\begin{gathered} V_x=\sqrt{{\left(70\mathrm{km}/\mathrm{h}\right)}^2-{\left(20.0\mathrm{km}/\mathrm{h}\right)}^2} \\ =67.08\mathrm{km}/\mathrm{h} \end{gathered}$$

The speed of aircraft relative to the ground is 67.08 km/h .