Relative velocity in two dimensions

Relative Velocity:

We encounter occasions where one or more objects move in a frame which is non-stationary with respect to another observer. For example, a boat crosses a river that is flowing at some rate or an aeroplane encountering wind during its motion. In all such instances, in order to describe the complete motion of the object, we need to consider the effect that the medium is causing on the object. While doing so, we calculate the relative velocity of the object considering the velocity of the particle as well as the velocity of the medium. Here, we will learn how to calculate the relative velocity.

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Examples of relative VelocityRelative Velocity Problems

Let us consider two objects, A and B moving with velocities V_a and V_b with respect to a common stationary frame of reference, say the ground, a bridge or a fixed platform.

The velocity of the object A relative to the object B can be given as,

Similarly, the velocity of the object B relative to that of object a is given by,

From the above two expressions, we can see that

Although the magnitude of both the relative velocities is equal to each other. Mathematically,

Examples of relative Velocity

We can understand the concept of relative velocity more clearly with the help of the following example.

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

Given, the velocity of the wind = V_w = 25 km/hr

The velocity of the plane = V_a= 100 km/hr

The relative 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

R²= 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,

Relative Velocity Problems

1) What is relative velocity?

Answer:

Relative velocity is defined as the velocity of an object B in the rest frame of another object A.

2) A motorcycle travelling on the highway at a velocity of 120 km/h passes a car travelling at a velocity of 90 km/h. From the point of view of a passenger on the car, what is the velocity of the motorcycle?

Solution:

Let us represent the velocity of the motorcycle as V_A and the velocity of the car as V_B.

Now, the velocity of the motorcycle relative to the point of view of a passenger is given as

V_AB = V_A – V_B

Substituting the values in the above equation, we get

V_AB = 120 km/h – 90 km/h = 30 km/h

Hence, the velocity of the motorcycle relative to the passenger of the car is 30 km/h.

3) A swimmer swimming across a river flowing at a velocity of 4 m/s swims at the velocity of 2 m/s. Calculate the actual velocity of the swimmer and the angle.

Solution:

The actual velocity of the swimmer can be found out as follows:

The angle is calculated as follows:

4) A person in an enclosed train car, moving at a constant velocity, throws a ball straight up into the air in her reference frame.

• At what point, does the ball land?
• Where does the ball land if the car slows down?
• Where does the ball land if the car speeds up?
• Where does it land if the car rounds a turn?

Solution:

1. The ball lands at the point from which it was thrown, i.e. back to the thrower’s hand.
2. It lands in front of the point from which it was thrown.
3. It lands behind the point from which it was thrown.
4. The ball will land to the left of the point from which it was thrown.

5) An aeroplane flies with a velocity of 450 m/s to the north, while an aeroplane B travels at a velocity of 500 m/s to the south beside aeroplane A. Calculate the relative velocity of the aeroplane A with respect to aeroplane B.
Solution:

The relative velocity of aeroplane A with respect to the velocity of aeroplane B is calculated as follows:

V_AB = V_A – V_B

Substituting the values in the equation, we get

V_AB = 450 m/s – (–500 m/s) = 950 m/s

The velocity of aeroplane B is considered negative, as it flies in the opposite direction to the of aeroplane A.

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