How to Calculate Relative Velocity in Same and Opposite Directions
Relative velocity is a fundamental concept in Physics, especially in kinematics, that describes how fast and in what direction one object is moving relative to another. It is particularly important when two or more objects are in motion and helps to analyze their behavior from different frames of reference. This concept applies to scenarios such as cars on highways, trains, boats in rivers, and even satellites in orbit.
In simple words, the relative velocity of an object is its velocity as measured or observed from another moving object. If two objects, A and B, are moving along the same straight line (say, the x-axis), their relative velocity helps determine how quickly they move towards or away from each other.
Definition and Derivation of Relative Velocity
Suppose objects A and B have initial positions \(x_A(0)\) and \(x_B(0)\), and velocities \(v_A\) and \(v_B\). After some time \(t\):
- Position of A: \(x_A(t) = x_A(0) + v_A t\)
- Position of B: \(x_B(t) = x_B(0) + v_B t\)
The displacement from A to B at any time is:
\( x_{BA}(t) = x_B(t) - x_A(t) = [x_B(0) - x_A(0)] + (v_B - v_A)t \)
Thus, the velocity of B relative to A is \(v_{BA} = v_B - v_A\).
Similarly, the velocity of A relative to B is \(v_{AB} = v_A - v_B = -v_{BA}\).
Key Formulas for Relative Velocity
| Situation | Formula | Description |
|---|---|---|
| Same direction | \(v_{BA} = v_B - v_A\) | Difference of velocities |
| Opposite direction | \(v_{BA} = v_B + v_A\) | Sum of speeds (when signs are opposite) |
| General (vector) | \(\vec{v}_{BA} = \vec{v}_B - \vec{v}_A\) | Applicable in 2D/3D motion |
Practical Steps: Solving a Relative Velocity Problem
- Choose the frame of reference (decide which object is the observer).
- Assign directions and signs to all velocities.
- Use the correct relative velocity formula as per their direction.
- Insert the numerical values and solve.
- Interpret the result in context (e.g., how quickly one object approaches or recedes from the other).
Special Cases and Applications
- If two objects have the same velocity and direction (\(v_A = v_B\)), their relative velocity is zero. They remain at a constant distance from each other.
- If they move exactly opposite (\(v_A\) positive, \(v_B\) negative), relative velocity is the sum of their speeds.
- For overtaking problems or trains passing each other, always pay attention to the direction and sign conventions.
Relative Velocity in 2D (Two Dimensions)
Sometimes, objects move in two perpendicular directions (for example, a boat crossing a river while the river flows sideways). In such cases, velocities are combined as vectors:
\( \vec{v}_{BA} = \vec{v}_B - \vec{v}_A \)
Use the parallelogram law of vector addition or resolve into components to find magnitude and direction.
Solved Examples
| Example | Solution/Explanation |
|---|---|
| Two trains on parallel tracks: Train A moves north at \(54\,\text{kmph}\), Train B south at \(90\,\text{kmph}\). |
Assigning directions: \(v_A = +15\,\text{m/s}\), \(v_B = -25\,\text{m/s}\); Relative velocity \(v_{BA} = v_B - v_A = -40\,\text{m/s}\). This means from A's perspective, B appears to move at \(40\,\text{m/s}\) southwards. |
|
A bus travels north at \(40\,\text{m/s}\), another bus travels south at \(60\,\text{m/s}\). Find relative velocity of A with respect to B. |
B's velocity is in the opposite direction, so \(v_{AB} = 40 - (-60) = 100\,\text{m/s}\). |
| Two cars start \(900\,\text{m}\) apart and move towards each other at \(1\,\text{m/s}\) and \(2\,\text{m/s}\). |
Relative velocity \(v_{21} = v_2 - v_1 = 2 - (-1) = 3\,\text{m/s}\). Time to meet \(= 900 / 3 = 300\,\text{s} = 5\,\text{min}\). |
Summary Table: Velocity vs Relative Velocity
| Aspect | Ordinary Velocity | Relative Velocity |
|---|---|---|
| Reference | Measured from fixed frame (usually ground) | Measured from another moving object |
| Formula | --- | \(v_{AB} = v_A - v_B\) |
| Nature | Absolute | Comparative |
Practice Approach
- Practice problems using different directions and speeds (e.g. trains, river boat, cars).
- Compare answers for both same and opposite directions to notice how the formula changes.
- Try applying vector methods for 2D (like boat in river) for advanced understanding.
Further Learning and Resources
- Revise related topics like Velocity Vectors and Velocity.
- For comparisons and more problem types, see Relative Speed.
- Explore additional solved examples and interactive questions at Relative Velocity.
Next Steps
- Practice more numerical and application-based problems for better conceptual clarity.
- Understand sign conventions and try different reference frames to master approach.
- Review examples of everyday motion for practical insight.
Mastering relative velocity equips you to confidently solve a variety of real-life and exam-based questions in Physics. Continue reinforcing your understanding with Vedantu’s interactive resources and topic guides.
FAQs on Relative Velocity in Physics: Concept, Formula & Examples
1. What is relative velocity in Physics?
Relative velocity refers to the velocity of one object as observed from another moving object’s frame of reference. It is a vector quantity that considers both magnitude and direction, and is essential for solving motion problems involving more than one moving body.
2. How do you calculate relative velocity when two objects move in the same direction?
When two objects move in the same direction along a straight line, the relative velocity of object A with respect to object B is calculated by subtracting their velocities:
VAB = VA − VB
3. How do you calculate relative velocity when two objects move in opposite directions?
For opposite direction motion, the relative velocity of object A with respect to object B is the sum of the magnitudes:
VAB = VA + VB
4. What is the formula for relative velocity in vector form?
The vector formula for relative velocity between two objects A and B is:
VAB = VA − VB
This formula applies for motion in any direction (1D, 2D, or 3D).
5. What is the difference between relative speed and relative velocity?
Relative speed is the magnitude of the relative velocity and does not consider direction (scalar), while relative velocity considers direction (vector).
- Relative velocity describes how fast and in which direction one object moves relative to another.
- Relative speed gives only the rate without direction.
6. Why do we calculate relative velocity?
We calculate relative velocity to determine how fast and in what direction one object appears to move when observed from another moving object. This concept is essential for analyzing motion in problems such as trains passing, boats in flowing rivers, and aircraft in the wind.
7. What are the steps to solve numerical problems on relative velocity?
To solve relative velocity numericals effectively:
1. Identify and mark the direction of each velocity.
2. Assign correct sign conventions (positive or negative).
3. Use the correct formula based on motion direction.
4. Substitute values and calculate.
5. Interpret the answer with correct units and direction.
8. Can you give an example of a relative velocity problem and its solution?
Example: Two trains move towards each other with speeds of 60 km/h and 40 km/h. What is the relative velocity of one train as seen from the other?
Solution: Relative velocity, VAB = VA + VB = 60 km/h + 40 km/h = 100 km/h. The trains approach each other at 100 km/h relative speed.
9. What is the significance of sign convention in relative velocity problems?
Sign convention is crucial because relative velocity calculations depend on direction. Choose a positive direction (e.g., right or north), assign signs to velocities accordingly, and apply formulas consistently—this avoids errors in final answers, especially in vector or 2D problems.
10. How is relative velocity used in boat and river problems?
In boat and river motion problems:
- The relative velocity of the boat with respect to the ground (or river bank) is the vector sum of the boat’s velocity in still water and the velocity of the river current.
- This helps determine the actual path, speed, and time taken to cross the river or reach a point.
11. Can relative velocity be zero? Under what condition?
Yes, relative velocity can be zero if two objects move in the same direction with equal velocity. In this case, they appear stationary to each other, as their positions do not change relative to one another over time.
12. What are some common mistakes students make when solving relative velocity questions?
Common mistakes include:
- Forgetting correct sign conventions for direction
- Confusing relative velocity with absolute velocity
- Using wrong formulas for same/opposite direction
- Ignoring vector nature in multi-dimensional problems
Tip: Always draw a diagram, choose a consistent reference frame, and double-check calculations.
