Relative Velocity Formula – Formula, Derivation, Applications, Example Problems

What is Relativity Velocity Formula?

Relative velocity is a key concept in physics that describes the velocity of an object or observer A in relation to the velocity of another object or observer B. The relative velocity formula helps in understanding how fast two objects are moving with respect to each other. The formula for relative velocity is expressed as

• Vₐᵦ is the velocity of object A relative to object B.
• Vₐ is the velocity of object A relative to a fixed point.
• Vᵦ is the velocity of object B relative to the same fixed point.

In simpler terms, to find out how fast two objects are moving towards or away from each other, one can subtract the velocity vector of one object from the other.

The concept of relative velocity was developed as part of classical mechanics, with significant contributions from Sir Isaac Newton. While Newton did not directly coin the term “relative velocity,” his laws of motion and gravity laid the groundwork for understanding motion in a relative sense. The relative velocity formula allows us to solve problems where objects are moving in different frames of reference, making it a fundamental equation in the study of motion.

Derivation of Relative Velocity Formula

The derivation of the relative velocity formula begins by considering the concept of velocity itself, which is a vector quantity. This means velocity has both magnitude and direction. When discussing relative velocity, we are dealing with the velocity of one object as observed from another moving object, rather than from a fixed point.

Let’s denote the velocity of object A as Vₐ and the velocity of object B as Vᵦ, both measured relative to a common fixed reference point. The goal is to find the velocity of object A as observed from object B, labeled as V_ₐᵦ. To derive the relative velocity formula, we follow these steps:

Consider Object A’s Velocity (V_ₐ): This is the speed and direction of A as measured from a fixed point.

Consider Object B’s Velocity (Vᵦ): Similarly, this is the speed and direction of B as measured from the same fixed point.

Subtract Object B’s Velocity from Object A’s Velocity: To find how fast A appears to be moving relative to B, we subtract the velocity of B from A. The formula used is:

This subtraction of vectors accounts for both the magnitude and direction of velocities.

Result Interpretation: The resulting vector Vₐᵦ represents the velocity of object A as it appears from object B. If both objects were moving towards each other, their relative speeds would add up.

Relative Velocity Formula for Same and Opposite Directions

Relative Velocity in the Same Direction

When two objects, A and B, are moving in the same direction, their relative velocity is calculated by subtracting the velocity of one object from the other. This is because both objects are moving along parallel paths, and we are interested in their speed difference. The formula is given by:

• Formula:

• V_ₐᵦ is the velocity of object A relative to object B.
• Vₐ​ is the velocity of object A.
• V_ᵦ ​ is the velocity of object B.

In this configuration, if Vₐ​​ is greater than V_ᵦ ​, the relative velocity V_ₐᵦ​ is positive, indicating that A is moving ahead of B. Conversely, if Vₐ​​ is less than V_ᵦ​, V_ₐᵦ is negative, meaning A is lagging behind B.

Relative Velocity in the Opposite Direction

When objects move in opposite directions, their relative velocity is calculated by adding their speeds. This scenario reflects the fact that both objects are contributing to the distance changing between them, effectively increasing the rate at which they separate or approach each other.

• Formula:

• V_ₐᵦ​ is the velocity of object A relative to object B.
• Vₐ is the velocity of object A moving in one direction.
• Vᵦ​ is the velocity of object B moving in the opposite direction.

Here, V_ₐᵦ​ will always be a positive value, representing the total rate of separation or approach, depending on the context.

Applications of Relativity Velocity Formula

1. Traffic Analysis: Helps in calculating the relative speeds of vehicles on roads to assess safety measures and traffic flow.
2. Aeronautics: Pilots use relative velocity to determine the speed and direction of other aircraft to avoid collisions and navigate effectively.
3. Sports: Analysts apply this formula to understand the motion of players relative to each other, improving strategies and training methods.
4. Astronomy: Scientists calculate the relative velocities of celestial bodies to predict orbits and interactions in space.
5. Maritime Navigation: Enables captains to determine the speed of other ships relative to their own, crucial for safe maneuvering and docking.
6. Physics Education: Teachers use examples of relative velocity to explain basic and advanced concepts in mechanics.
7. Robotics: Engineers design robots that use relative velocity calculations to interact with moving objects and environments.

Example Problems on Relative Velocity Formula

Problem 1: Cars on the Highway

Question: Car A is traveling at 80 km/h east, and Car B is traveling at 60 km/h east. What is the velocity of Car A relative to Car B?

Solution:

Use the formula for relative velocity in the same direction: Vₐᵦ = V_ₐ + V_ᵦ

Plug in the values:

𝑉_ₐᵦ = 80 km/h − 60 km/h=20 km/h

Answer: The velocity of Car A relative to Car B is 20 km/h east.

Problem 2: Approaching Trains

Question: Train A moves west at 70 km/h, and Train B moves east at 80 km/h. What is the relative velocity of Train A with respect to Train B?

Solution:

Use the formula for relative velocity in opposite directions: Vₐᵦ = V_ₐ + V_ᵦ​

Since they are moving towards each other:

𝑉_ₐᵦ = 70 km/h + 80 km/h = 150 km/h

Answer: The relative velocity of Train A with respect to Train B is 150 km/h.

Problem 3: Crossing Boats

Question: Boat A is moving north at 10 m/s and Boat B is moving west at 15 m/s. What is the relative velocity of Boat A with respect to Boat B?

Solution:

Since the boats are moving perpendicular to each other, use vector subtraction to find the relative velocity:

𝑉⃗_ₐᵦ = 𝑉⃗ₐ − 𝑉⃗_ᵦ​

Represent the velocities as vectors:

𝑉⃗ₐ = 10 m/s 𝑖, 𝑉⃗_ᵦ = 15 m/s 𝑗​

Subtract the vectors:

𝑉⃗ₐᵦ = (10 m/s 𝑖) − (−15 m/s 𝑗) = 10 m/s 𝑖 + 15 m/s 𝑗

Calculate the magnitude of 𝑉⃗_ₐᵦ ​:

∣𝑉⃗ₐᵦ∣ = √(10² + 15²) m/s = √100+225 m/s = √325 m/s ≈ 18.03 m/s

Answer: The relative velocity of Boat A with respect to Boat B is approximately 18.03 m/s in a northeast direction.

FAQs

What is the Formula for Relativistic Velocity?

The relativistic velocity formula combines velocities using Lorentz transformations, crucial for velocities approaching the speed of light.

Why Do We Calculate Relative Velocity?

Calculating relative velocity helps predict the motion of objects as observed from different reference frames, enhancing safety and planning.

What is Relative Velocity for Dummies?

Relative velocity measures how fast one object moves in relation to another, simplifying understanding of motion interactions.