Relative Velocity in a Plane | Physics Class 11 - NEET PDF Download

What is Relative Motion Velocity?

Relative velocity is the velocity of an object as seen from another moving object (or reference frame). This observed velocity depends on the chosen frame of reference: the other object may be at rest, moving with the same velocity, moving slower, moving faster, or even moving in a different direction. The idea of relative velocity applies to motion in a straight line and can be easily extended to motion in a plane or in three dimensions.

Velocity

Velocity is the rate of change of displacement with time. It is a vector quantity having both magnitude and direction. The magnitude of velocity is the speed. For a two-dimensional velocity with components v_x and v_y.

|v| = v = √(v_x² + v_y²)

Acceleration

Acceleration is the rate of change of velocity with respect to time. In component form,  it can be presented as –

Relative Motion Velocity in Two Dimensions

Consider two objects A and B moving with velocity vectors V_a and V_b as measured in a common inertial frame (for example, the ground). The velocity of A as seen from B (velocity of A relative to B) is given by vector subtraction:

V_ab = V_a - V_b

Similarly, the velocity of B as seen from A is

V_ba = V_b - V_a = -V_ab

Therefore, the magnitudes satisfy

|V_ab| = |V_ba|

If V_a = V_b, then V_ab = V_ba = 0 and the two objects are at rest relative to each other.

Component form

Writing the vectors in components,

V_a = (V_ax, V_ay), V_b = (V_bx, V_by)

Then

V_ab = (V_ax - V_bx, V_ay - V_by)

The magnitude of V_ab is

|V_ab| = √[(V_ax - V_bx)² + (V_ay - V_by)²]

Interpretation and special cases

• If the two velocity vectors are in the same direction (components have same sign), the relative speed may be small; one object may appear slow or nearly at rest relative to the other.
• If they are in opposite directions, the relative speed is larger than either individual speed; each appears to approach the other more rapidly.
• Relative velocity depends on the chosen frame. Transforming from one inertial frame to another uses the same subtraction formula: measured velocity in new frame = velocity in old frame minus velocity of new frame relative to old frame.

Graphical method (vector diagram)

To find relative velocity graphically, place the tail of V_b at the tail of V_a and draw V_ab as the vector from the tip of V_b to the tip of V_a. Equivalently, V_ab = V_a + (-V_b).

Solved Questions

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:

Choose a coordinate system with +x towards east and +y upwards.

The velocity of rain relative to ground is V_r = (0, -35) m/s.

The velocity of the bicycle relative to ground is V_b = (-12, 0) m/s (east to west is negative x).

Relative velocity of rain with respect to bicycle is

V_rb = V_r - V_b

V_rb = (0 - (-12), -35 - 0) = (12, -35) m/s

The magnitude is

|V_rb| = √(12² + 35²) = √(144 + 1225) = √1369 = 37 m/s

The direction relative to the vertical: the horizontal component is 12 m/s towards the east (positive x), and the vertical component is 35 m/s downwards. The angle θ measured from the vertical towards the east satisfies

tan θ = (horizontal component)/(vertical component) = 12/35

θ = arctan(12/35) ≈ 19°

Thus the rain appears to come at 37 m/s along a line inclined 19° from the vertical towards the east (equivalently, the person should tilt the umbrella 19° from the vertical towards the west to keep the rain off).

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:

Take +x towards east and +y towards north. The plane's velocity relative to air (its airspeed) is V_p = (0, -100) km/hr (southward). The wind velocity relative to ground is V_w = (-25, 0) km/hr (westward).

The resultant velocity of the plane relative to ground is

V = V_p + V_w = (-25, -100) km/hr

The magnitude is

|V| = √((-25)² + (-100)²) = √(625 + 10000) = √10625 = 103.077 km/hr (approximately)

The direction relative to the southward direction: the horizontal component is 25 km/hr towards west and the southward component is 100 km/hr. The angle θ west of south satisfies

tan θ = 25/100 = 1/4

θ = arctan(1/4) ≈ 14.0°

So the plane's ground track is 103.08 km/hr at 14.0° west of south.