The theory of relativity deals with the relative motion of an object to another frame of reference. It describes the motion between two different frames interacting with each other.

**Relative velocity is one of the terms which describes the relative motion between two bodies. It is the velocity of an object observed by an observer in another frame, either in a stationary or moving frame. The relative velocity of a plane is concerned with the motion of a plane in the air measured by an observer in another frame.**

In this post, you will learn about the relative velocity of a plane in various aspects.

**What is relative velocity of a plane?**

We are all very fond of watching the plane from the ground in our childhood. The speed at which the plane is moving is observed by us. The plane is moving and is in a different frame of reference, and we are in the ground on rest state.

**The relative velocity of a plane is defined as “”the velocity of a plane moving in air is measured by an observer in another frame either stationary or moving with same or varying velocity.”” The motion of the plane in the air is influenced by the various aero-dynamical factors, which may or may not affect the velocity of a plane.**

The relative velocity of a plane is associated with the airspeed or wind speed. The relative velocity of a plane measured in air is different from that measured on ground. This statement means that if you measure the velocity of the plane in the air medium, the measured value of velocity is different from the value obtained from an observer who measured the velocity of the plane from the ground.

**This indeed tells that velocity is not accurate; it is relative to the frame and the observer observing the event. The interesting fact is that relative velocity of the plane for the passenger sitting inside the plane is zero as they are in the same reference frame.**

**How to find relative velocity of a plane?**

In order to find the relative velocity of a plane with respect to another frame of reference, let us consider an example of an observer on the ground who measures the velocity of a plane flying in the air.

**Since the plane is flying in the air, there will be an influence of airspeed on the plane’s velocity so that the speed of the air on the plane is necessary to understand. Using plane’s velocity in air and the speed of the air on the ground, the velocity of the plane measured on ground can easily be calculated.**

Let v_{PG} be the velocity of the plane measured on ground and v_{PA} be the plane’s velocity in air medium, and v_{GA} be airspeed on ground. Using the law of vector addition, the velocity of the plane relative to the ground is given by

v_{PG} = v_{PA}+v_{GA}

Here we took the velocity of air on the ground because the observer on the ground is stationary, and only there is the motion of air common in both the frame. Thus, the speed of air on the ground is helpful to find the relative velocity of a plane on the ground.

**Suppose the plane is moving with the velocity of 250 miles per hour with respect to air and the speed of the air on the ground is 80 miles per hour, blowing vertically in the east, making an angle of 60°, then the relative velocity of the plane with respect to the ground can be written as follows.**

From the above formula, we know that the relative velocity of a plane with respect to the ground is

v_{PG} = v_{PA}+v_{GA}

The magnitude of the velocity of the plane relative to the ground is given as

[latex]v_P_G=v_P_G\hat{i}+0\hat{j}[/latex]

Where; i and j are the unit vector. The unit vector j is 0 because the components of j along the ground is zero.

The magnitude of the velocity of the air on the ground is given as

[latex]v_G_A=80cos(60)\hat{i}+80sin(60)\hat{j}[/latex]

On solving the above equation we will get

[latex]v_G_A=40\hat{i}+40\sqrt{3}\hat{j}[/latex]

Because [latex]cos(60)=\frac{1}{2}[/latex] and [latex]sin(60)=\frac{\sqrt{3}}{2}[/latex]

The magnitude of relative velocity of a plane in air is given by

[latex]v_P_A=v_x\hat{i}+v_y\hat{j}[/latex]

On solving for the y-components, we get

[latex]v_y+40\sqrt{3}=0[/latex]

[latex]v_y=-40\sqrt{3}[/latex]

Here negative sign indicates that the plane’s motion is in the opposite direction to the blowing air.

And for c-components, we get

[latex]v_x+40=v_P_G[/latex]

The magnitude of any two components is given by [latex]\sqrt{v_x^2+v_y^2}[/latex]

Using this equation, we can solve for x-components; since we know the value of the velocity of the plane with respect to air.

[latex]v_P_A=\sqrt{v_x^2+v_y^2}[/latex]

Substituting the value of v_{PA} and v_{Y} and then rearranging we get v_{x }as

[latex]v_x=\sqrt{250^2-(40\sqrt{3})^2}[/latex]

[latex]v_x= \sqrt{62500-4800}[/latex]

v_{x} = 240.2 mph

The direction of the vector is given by

[latex]tan\phi=\frac{v_y}{v_x}[/latex]

[latex]tan\phi=\frac{40\sqrt{3}}{240.2}[/latex]

[latex]\phi=\tan^{-1}(0.288)[/latex]

Φ=16.06°.

This gives the velocity of a plane with respect to the ground along with magnitude and direction as velocity is a vector quantity.

**What is the formula to get the relative velocity of a plane?**

The motion of the plane in the air is related to the motion of air and the observer observing on the ground.

**If we need to find the relative velocity of a plane on the ground, we must consider the plane’s velocity in the air and air velocity on the ground. Let v _{AB} is the velocity of the plane relative to the ground, v_{BC} is the velocity of air in the ground, and v_{AC} is the velocity of the plane in the air, then the formula will be**

**v _{AB}=v_{BC}+v_{AC}**

If the velocity of the plane is to be measured with respect to air only, then let v_{A} be the velocity of the air and v_{B} be the velocity of the plane, then the formula will be

**v _{AB} = v_{A}–v_{B}**

Both formulae are applicable to all types of relative velocity with respect to any frame of reference.

**Significance of relative velocity of a plane**

- Understanding the relative velocity of a plane explains why taking off of a plane and landing is done on different runways on different days.
**Both take-off and landing of a plane are associated with the wind’s direction, which requires lower ground speed to become airborne.** **The relative velocity of a plane plays a vital role in controlling air traffic.**- The plane’s velocity in the air is called airspeed, the velocity of air on the ground is called wind speed, and the velocity of the plane relative to the ground is called ground speed,
**which helps us understand the functioning of wind tunnels and the flight of kites.**

**Frequently Asked questions**

**What is the role of relative velocity in controlling air traffic?**

Since relative velocity specifies the direction of the wind, it can control the air traffic.

**The generation of aero-dynamical lift, which specifies the plane’s motion, corresponds to the relative velocity of plane and air. However, we cannot calculate the airspeed directly so that we can calculate the relative velocity of the plane with respect to the ground, which helps to find the airspeed. This airspeed describes the plane’s velocity in an air medium and can help the pilot take control of their plane so that air traffic can be controlled.**

**Does the relative velocity change with distance?**

No, relative velocity does not change; it remains the same as long as you observe the motion from the same frame.

**The relative velocity is concerned with only the observation. For example, you are measuring the velocity of a bike moving with a certain velocity from a car that is one meter away from the bike. Even when the bike moves faster and is moving 2m away from you, you will get the relative velocity the same because you have covered some distance relative to that bike.**

**What is the importance of relative velocity?**

Relative velocity is important to understand the interaction between the two objects in the different frames.

**Most essentially, relative velocity helps us to understand whether the object is under rest or motion. The concept of relative velocity helps us to calculate the velocity of stars and asteroids relative to the earth. It is also helpful in the process of rocket launching and determining the speed.**