Relative Velocity in Two Dimensions

The concept of relative motion velocity in a plane is quite similar to the whole concept of relative velocity in a straight line. Considering various occasions, we take in more than one object move in a frame which is non-stationary in respect to another viewer.

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What is Relative Motion Velocity?

The relative motion velocity refers to an object which is relative to some other object that might be stationary, moving with the same velocity, or moving slowly, moving with higher velocity or moving in the opposite direction. The wide concept of relative velocity can be very easily extended for motion along a straight line, to include motion in a plane or either in three dimensions.

Velocity

Velocity refers to a physical vector quantity which is described by both magnitude and direction. The magnitude or scalar absolute value of velocity is referred to as speed. As stated by the Pythagorean Theorem, the magnitude of the velocity vector is given by –

| v | = v = √ ( v_x ²+ v_y ² )

Browse more Topics under Motion In A Plane

• Introduction to Motion in a Plane
• Scalars and Vectors
• Resolution of Vectors and Vector Addition
• Addition and Subtraction of Vectors – Graphical Method
• Uniform Circular Motion
• Projectile Motion

You can download Motion in a Plane Cheat Sheet by clicking on the download button below

Acceleration

Acceleration is the rate of change of velocity of an object with respect to time. Numerically or in terms of components, it can be presented as –

a_x =   \( \frac{d}{dt} \)   v_x

a_y =    \( \frac{d}{dt} \)   v_y

Relative Motion Velocity in Two Dimensions

Let us consider two objects A and B who are moving with the velocity at Va and Vb with respect to some common frame of reference, for example, may be the ground or the earth. In this case, the velocity of object A relative to that of B will be –

V_ab=  V_a –  V_b

Similarly, the velocity of object B relative to that of A will be :

V_ba = V_b– V_a

Therefore,

V_ba= – V_ba

And | V_ab | = | V_ba |

When two objects seem to be stationary for one another, in that case –

V_b = V_a

V_ba = V_ab = 0

The magnitude of V_ba and V_ab will be lower than the magnitude of V_a and V_b if V_a and V_b are of same sign. Object A appears faster to B and B appears slower to A if V_a > V_b.

The magnitude of V_ba and V_ab will be higher than the magnitude of V_a and V_b if V_a and V_b are of opposite sign. In this case, both objects will appear moving faster to one another.

Solved Questions for You

Q1: If rain is falling vertically at a speed of 35 m/s and a person is riding a bicycle at 12 m/s ( east to west ) then the relative motion velocity of rain will be V_rb

Solution : V_rb = V_r – V_b = ( 35 – 12 ) m/s = 23 m/s
Tan θ =  V_w /  V_r = 12 / 35 = 0.343
θ = 19 degree
The person must hold an umbrella at an angle of 19 degrees with vertical towards the west to avoid the rain.

Q2. A plane is traveling at velocity 100 km/hr, in the southward direction. It encounters wind traveling in the west direction at a rate of 25 km/hr. Calculate the resultant velocity of the plane.

Solution: Given, the velocity of the wind = V_w = 25km / hr
The velocity of the plane = V_a = 100 km / hr

The relative motion velocity of the plane with respect to the ground can be given as the angle between the velocity of the wind and that of the plane is 90°. Using the Pythagorean theorem, the resultant velocity can be calculated as,

R₂= (100 km/hr)²  + (25 km/hr)²
R₂ = 10 000 km² / hr² + 625 km² / hr²
= 10 625 km² / hr²
Hence, R = 103.077 km/hr

Using trigonometry, the angle made by the resultant velocity with respect to the horizontal plane can be given as,

Tan θ = ( window velocity / airplane velocity )
Tan θ =( 25/100 )
θ = tan ⁻¹(1/4)
θ =1/4 radians