The Relative Velocity Formula gives the velocity of one object as observed from another moving object, expressed as νAB = νA − νB for one-dimensional motion. This concept is a cornerstone of kinematics, covered in NCERT Class 11 Physics Chapter 3 (Motion in a Straight Line) and Chapter 4 (Motion in a Plane). It appears regularly in JEE Main, JEE Advanced, and NEET papers. This article covers the formula expression, variable table, derivation, a complete formula sheet, three solved examples of increasing difficulty, CBSE exam tips, common mistakes, and JEE/NEET applications.
Key Relative Velocity Formulas at a Glance
Quick reference for the most important relative velocity formulas used in CBSE and competitive exams.
- 1D same direction: \( v_{AB} = v_A – v_B \)
- 1D opposite directions: \( v_{AB} = v_A + v_B \)
- 2D vector form: \( \vec{v}_{AB} = \vec{v}_A – \vec{v}_B \)
- Magnitude (2D): \( |\vec{v}_{AB}| = \sqrt{v_A^2 + v_B^2 – 2v_A v_B \cos\theta} \)
- Rain-man problem: \( \vec{v}_{\text{rain,man}} = \vec{v}_{\text{rain}} – \vec{v}_{\text{man}} \)
- Relative displacement: \( s_{AB} = (v_A – v_B) \cdot t \)
- Time to meet: \( t = \dfrac{d}{v_A + v_B} \) (opposite directions)
What is the Relative Velocity Formula?
The Relative Velocity Formula defines the velocity of object A as measured by an observer on object B. In everyday life, all motion is relative. When you sit in a moving train, a person on the platform sees you moving fast. But a co-passenger sees you as stationary. This difference in observation is captured by the concept of relative velocity.
According to NCERT Class 11 Physics, Chapter 3, relative velocity is the vector difference of the velocities of two objects. If object A moves with velocity \( \vec{v}_A \) and object B moves with velocity \( \vec{v}_B \), both measured from the same ground frame, then the velocity of A relative to B is \( \vec{v}_{AB} = \vec{v}_A – \vec{v}_B \).
This formula applies to one-dimensional motion (along a straight line) and two-dimensional motion (in a plane). In 1D, the sign of velocity indicates direction. In 2D, vector subtraction uses the parallelogram law or component method. The concept extends to problems involving rivers, rain, aircraft, and collisions. It is fundamental to understanding reference frames in classical mechanics.
The Relative Velocity Formula is not just a theoretical tool. It has direct applications in navigation, aviation, sports science, and traffic engineering.
Relative Velocity Formula — Expression and Variables
One-Dimensional Motion
When two objects A and B move along the same straight line:
\[ v_{AB} = v_A – v_B \]
Here, \( v_{AB} \) is the velocity of A with respect to B. Similarly, the velocity of B with respect to A is:
\[ v_{BA} = v_B – v_A \]
Note that \( v_{AB} = -v_{BA} \). The two relative velocities are equal in magnitude but opposite in direction.
Two-Dimensional (Vector) Form
\[ \vec{v}_{AB} = \vec{v}_A – \vec{v}_B \]
The magnitude of the relative velocity when the angle between \( \vec{v}_A \) and \( \vec{v}_B \) is \( \theta \) is:
\[ |\vec{v}_{AB}| = \sqrt{v_A^2 + v_B^2 – 2v_A v_B \cos\theta} \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| \( v_{AB} \) | Velocity of A relative to B | m/s |
| \( v_A \) | Velocity of object A (ground frame) | m/s |
| \( v_B \) | Velocity of object B (ground frame) | m/s |
| \( \vec{v}_{AB} \) | Relative velocity vector of A w.r.t. B | m/s |
| \( \theta \) | Angle between velocity vectors of A and B | degrees / radians |
| \( |\vec{v}_{AB}| \) | Magnitude of relative velocity | m/s |
| \( t \) | Time elapsed | s |
| \( s_{AB} \) | Relative displacement of A w.r.t. B | m |
Derivation
Consider two objects A and B moving along the x-axis. At time \( t = 0 \), their positions are \( x_A(0) \) and \( x_B(0) \). After time \( t \), their positions become:
\[ x_A(t) = x_A(0) + v_A t \quad \text{and} \quad x_B(t) = x_B(0) + v_B t \]
The position of A relative to B is \( x_{AB}(t) = x_A(t) – x_B(t) \). Differentiating with respect to time gives the relative velocity:
\[ v_{AB} = \frac{d}{dt}[x_A(t) – x_B(t)] = v_A – v_B \]
This derivation confirms that relative velocity is simply the time derivative of relative displacement. The result holds in all inertial (non-accelerating) reference frames in classical mechanics.
Complete Kinematics & Relative Motion Formula Sheet
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Relative Velocity (1D, same direction) | \( v_{AB} = v_A – v_B \) | vA, vB = velocities of A and B | m/s | Class 11, Ch 3 |
| Relative Velocity (1D, opposite directions) | \( v_{AB} = v_A + v_B \) | vA, vB = speeds of A and B | m/s | Class 11, Ch 3 |
| Relative Velocity (2D vector form) | \( \vec{v}_{AB} = \vec{v}_A – \vec{v}_B \) | Vector velocities of A and B | m/s | Class 11, Ch 4 |
| Magnitude of Relative Velocity (2D) | \( |\vec{v}_{AB}| = \sqrt{v_A^2 + v_B^2 – 2v_A v_B \cos\theta} \) | θ = angle between velocity vectors | m/s | Class 11, Ch 4 |
| Relative Displacement | \( s_{AB} = (v_A – v_B)\,t \) | t = time elapsed | m | Class 11, Ch 3 |
| Time to Overtake | \( t = \dfrac{d}{v_A – v_B} \) | d = initial separation, vA > vB | s | Class 11, Ch 3 |
| Time to Meet (opposite directions) | \( t = \dfrac{d}{v_A + v_B} \) | d = initial separation | s | Class 11, Ch 3 |
| River-Boat: Resultant Velocity | \( v_r = \sqrt{v_b^2 + v_w^2} \) | vb = boat speed, vw = water speed (perpendicular) | m/s | Class 11, Ch 4 |
| River-Boat: Drift | \( x = v_w \cdot t \) | t = time to cross river | m | Class 11, Ch 4 |
| Rain-Man Problem | \( \vec{v}_{\text{rain,man}} = \vec{v}_{\text{rain}} – \vec{v}_{\text{man}} \) | Vector subtraction in 2D | m/s | Class 11, Ch 4 |
| Angle of Umbrella (Rain-Man) | \( \tan\phi = \dfrac{v_{\text{man}}}{v_{\text{rain}}} \) | φ = angle from vertical | degrees | Class 11, Ch 4 |
Relative Velocity Formula — Solved Examples
Example 1 (Class 9–10 Level): Two Trains Moving in the Same Direction
Problem: Train A moves east at 60 m/s. Train B moves east at 40 m/s on a parallel track. Find the velocity of A relative to B and the velocity of B relative to A.
Given: \( v_A = +60 \) m/s (east), \( v_B = +40 \) m/s (east)
Step 1: Write the formula: \( v_{AB} = v_A – v_B \)
Step 2: Substitute values: \( v_{AB} = 60 – 40 = 20 \) m/s
Step 3: Find velocity of B relative to A: \( v_{BA} = v_B – v_A = 40 – 60 = -20 \) m/s
Step 4: Interpret the sign. Positive means east; negative means west. So B appears to move west at 20 m/s as seen from A.
Answer
Velocity of A relative to B = +20 m/s (east). Velocity of B relative to A = −20 m/s (west).
Example 2 (Class 11–12 Level): River-Boat Problem
Problem: A boat can travel at 5 m/s in still water. A river flows at 3 m/s due east. The boat heads due north (perpendicular to the river). Find (a) the resultant velocity of the boat and (b) the angle it makes with the north direction.
Given: Boat velocity w.r.t. water: \( v_b = 5 \) m/s (north), River velocity: \( v_w = 3 \) m/s (east)
Step 1: The velocity of the boat relative to the ground is the vector sum: \( \vec{v}_{\text{boat,ground}} = \vec{v}_{b} + \vec{v}_{w} \)
Step 2: Since the two velocity vectors are perpendicular, use Pythagoras:
\[ |\vec{v}_{\text{boat,ground}}| = \sqrt{v_b^2 + v_w^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83 \text{ m/s} \]
Step 3: Find the angle \( \phi \) east of north:
\[ \tan\phi = \frac{v_w}{v_b} = \frac{3}{5} = 0.6 \implies \phi = \tan^{-1}(0.6) \approx 30.96^\circ \]
Step 4: The boat drifts east due to the river current. Its actual path makes about 31° east of north.
Answer
Resultant speed ≈ 5.83 m/s, directed approximately 31° east of north.
Example 3 (JEE/NEET Level): Rain-Man Problem with Angle Change
Problem: Rain falls vertically at 10 m/s. A man walks east at 6 m/s. (a) At what angle should he hold his umbrella to stay dry? (b) If he doubles his speed to 12 m/s, what is the new angle?
Given: \( v_{\text{rain}} = 10 \) m/s (downward, i.e., −y direction), \( v_{\text{man,1}} = 6 \) m/s (east), \( v_{\text{man,2}} = 12 \) m/s (east)
Step 1: The velocity of rain relative to man is:
\[ \vec{v}_{\text{rain,man}} = \vec{v}_{\text{rain}} – \vec{v}_{\text{man}} \]
Step 2: Component form. \( \vec{v}_{\text{rain}} = (0,\,-10) \) m/s; \( \vec{v}_{\text{man,1}} = (6,\,0) \) m/s.
\[ \vec{v}_{\text{rain,man,1}} = (0 – 6,\; -10 – 0) = (-6,\,-10) \text{ m/s} \]
Step 3: Angle from vertical (downward direction):
\[ \tan\phi_1 = \frac{6}{10} = 0.6 \implies \phi_1 = \tan^{-1}(0.6) \approx 30.96^\circ \text{ (west of vertical)} \]
The man must tilt the umbrella about 31° toward the east (front) from vertical.
Step 4: For doubled speed, \( v_{\text{man,2}} = 12 \) m/s:
\[ \tan\phi_2 = \frac{12}{10} = 1.2 \implies \phi_2 = \tan^{-1}(1.2) \approx 50.19^\circ \]
Step 5: Notice the angle increases significantly. The faster the man walks, the more he must tilt the umbrella forward.
Answer
(a) Umbrella angle ≈ 31° from vertical (toward east). (b) At double speed, angle ≈ 50.2° from vertical.
CBSE Exam Tips 2025-26
- Always define a positive direction first. Assign one direction (usually east or right) as positive. Apply signs consistently throughout the problem. This single habit eliminates most sign errors.
- Memorise the two key 1D cases. Same direction: \( v_{AB} = v_A – v_B \). Opposite directions: \( v_{AB} = v_A + v_B \). In CBSE 3-mark questions, writing the correct formula earns 1 mark even if the calculation has a minor error.
- Draw a vector diagram for 2D problems. We recommend sketching the velocity vectors before subtracting. A clear diagram prevents confusion between \( \vec{v}_A – \vec{v}_B \) and \( \vec{v}_B – \vec{v}_A \).
- Use the cosine formula for non-perpendicular vectors. The formula \( |\vec{v}_{AB}| = \sqrt{v_A^2 + v_B^2 – 2v_A v_B \cos\theta} \) is directly applicable. Remember that \( \theta \) is the angle between the two velocity vectors, not the angle of relative velocity.
- State units in every step. CBSE examiners deduct marks for missing units. Write “m/s” after every velocity value in your solution steps.
- Practice rain-man and river-boat problems separately. These are the two most common 2D relative velocity problem types in CBSE Class 11 and Class 12 revision tests for 2025-26.
Common Mistakes to Avoid
- Mistake 1: Reversing the subtraction order. Students often write \( v_{AB} = v_B – v_A \) instead of \( v_A – v_B \). Remember: the subscript order tells you which velocity to subtract. “A relative to B” means \( v_A – v_B \).
- Mistake 2: Adding speeds instead of subtracting for same-direction motion. When two objects move in the same direction, the relative speed is the difference, not the sum. The sum applies only when they move toward each other (opposite directions).
- Mistake 3: Ignoring vector nature in 2D problems. In the river-boat problem, students sometimes add the boat speed and river speed algebraically. These are perpendicular vectors. Use the Pythagorean theorem or the cosine formula.
- Mistake 4: Confusing relative velocity with relative speed. Relative velocity is a vector (has direction). Relative speed is its magnitude (always positive). In CBSE answer writing, use the correct term.
- Mistake 5: Forgetting that \( v_{AB} = -v_{BA} \). Many students treat the two as independent. They are equal in magnitude and opposite in direction. If you find one, the other is simply its negative.
JEE/NEET Application of the Relative Velocity Formula
In our experience, JEE aspirants encounter the Relative Velocity Formula in at least 2–3 questions per paper, spread across kinematics and collision problems. NEET uses it primarily in kinematics and biomechanics contexts. Here are the three most common application patterns:
Pattern 1: Closest Approach and Collision Problems (JEE Main)
Two objects move with given velocities and initial separation. You must find the minimum distance between them or the time of collision. The key insight is to work in the reference frame of one object. In that frame, the other object moves with the relative velocity \( \vec{v}_{AB} \). The minimum distance is the perpendicular from the initial position of the “moving” object to its trajectory line in this frame. This reduces a 2D problem to a simple geometry problem.
Pattern 2: River-Boat Optimisation (JEE Advanced)
JEE Advanced frequently asks for the angle at which a boat should head to (a) cross the river in minimum time or (b) cross with minimum drift. For minimum time, the boat should head straight across (perpendicular to flow). For minimum drift, the boat should head upstream at an angle \( \sin^{-1}(v_w / v_b) \) from the perpendicular, provided \( v_b > v_w \). These results follow directly from the Relative Velocity Formula and optimisation using calculus or trigonometry.
Pattern 3: Relative Motion of Projectiles (JEE Advanced / NEET)
Two projectiles are launched simultaneously from different points. From the reference frame of one projectile, the other moves in a straight line (because both experience the same gravitational acceleration, which cancels in relative motion). This is a powerful result. It means: in the frame of one projectile, the relative velocity is constant and equal to the initial relative velocity. JEE Advanced uses this to set elegant “will they collide?” problems. Our experts suggest practising at least 10 such problems before the exam.
FAQs on Relative Velocity Formula
We hope this comprehensive guide on the Relative Velocity Formula has helped you build a strong conceptual and problem-solving foundation. For further reading, explore our related articles: Resultant Force Formula for vector addition concepts, Measurement Formulas for unit conversion techniques, and Physics Formulas Hub for the complete list of NCERT Class 11 and Class 12 formulas. For the official NCERT syllabus, refer to the NCERT official website. You may also find our articles on Kinematic Viscosity Formula and Dynamic Viscosity Formula useful for fluid mechanics topics in JEE.