---
title: "Relative velocity and acceleration"
subject: "Applied Mechanics (Dynamics)"
chapter: 1
key: "semester_3/applied_mechanics_dynamics:relative-velocity-acceleration-define"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/relative-velocity-acceleration-define
site: BCT Engineering
---

# Relative velocity and acceleration

Applied Mechanics (Dynamics) · Chapter 1: Curvilinear Motion of Particles

- Asked in: 2075 Chaitra, 2+2 marks
- Video: [Dynamics - Lesson 14: Relative Motion with Translating Axis](https://www.youtube.com/watch?v=yQNiowpFWaY) — Jeff Hanson
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/relative-velocity-acceleration-define
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-1

## Question

Define relative velocity and acceleration with suitable example.

## Short answer

Relative velocity of A with respect to B is the vector difference v_A/B = v_A - v_B; relative acceleration is similarly a_A/B = a_A - a_B. For example, car A east at 60 km/h and B north at 80 km/h gives v_A/B = 100 km/h at 53.1° south of east.

## Answer

#### Relative velocity
If two particles A and B have velocities $\vec v_A$ and $\vec v_B$ (both measured in a fixed frame), the **relative velocity of A with respect to B** is the velocity of A as seen by an observer moving with B:

$$
\vec v_{A/B}=\vec v_A-\vec v_B
$$

Equivalently, $\vec v_A=\vec v_B+\vec v_{A/B}$. It follows from the relative position $\vec r_{A/B}=\vec r_A-\vec r_B$ by differentiating.

#### Relative acceleration
In the same way, the **relative acceleration of A with respect to B** is

$$
\vec a_{A/B}=\vec a_A-\vec a_B
$$

(valid when the observer on B translates without rotating). Note that $\vec v_{B/A}=-\vec v_{A/B}$ and $\vec a_{B/A}=-\vec a_{A/B}$.

#### Example
Car A moves east at 60 km/h and car B moves north at 80 km/h. The velocity of A relative to B is

$$
\vec v_{A/B}=60\hat i-80\hat j\ \text{km/h},\qquad |v_{A/B}|=\sqrt{60^2+80^2}=100\ \text{km/h}
$$

at $\tan^{-1}(80/60)=53.1^\circ$ south of east. A passenger in B sees A moving at 100 km/h in that direction. If A accelerates at $2\hat i$ m/s$^2$ and B at $1\hat j$ m/s$^2$, then $\vec a_{A/B}=2\hat i-1\hat j$ m/s$^2$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the relative velocity of A with respect to B?
   A. v_A + v_B
   B. v_A - v_B, the velocity of A as seen by an observer moving with B
   C. v_B - v_A
   D. The average of v_A and v_B
   Answer: B. v_A/B = v_A - v_B, obtained by differentiating r_A/B = r_A - r_B. (Key point: Definition)

2. How are v_B/A and v_A/B related?
   A. v_B/A = v_A/B
   B. v_B/A = -v_A/B
   C. v_B/A = 2 v_A/B
   D. They are unrelated
   Answer: B. Swapping the observer reverses the sign. (Key point: Reverse relation)

3. When is a_A/B = a_A - a_B valid?
   A. When the observer on B translates without rotating
   B. Only when B is at rest
   C. Only when both move in a circle
   D. Always, even for rotating observers
   Answer: A. A rotating frame adds extra terms such as Coriolis acceleration. (Key point: Relative acceleration condition)

4. Car A moves east at 60 km/h and car B north at 80 km/h. What is |v_A/B|?
   A. 20 km/h
   B. 140 km/h
   C. 100 km/h
   D. 70 km/h
   Answer: C. v_A/B = 60i - 80j, magnitude sqrt(60^2 + 80^2) = 100 km/h. (Key point: Example)

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