---
title: "Radial and transverse acceleration derivation"
subject: "Applied Mechanics (Dynamics)"
chapter: 1
key: "semester_3/applied_mechanics_dynamics:radial-transverse-acceleration"
source: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/radial-transverse-acceleration
site: BCT Engineering
---

# Radial and transverse acceleration derivation

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

- Asked in: 2076 Chaitra, 6 marks; 2074 Chaitra, 4 marks (asked 2 times)
- Video: [Engineering Dynamics 14.1-04 Polar Acceleration Vector](https://www.youtube.com/watch?v=I64D1AXPkGU) — Brett Tallman
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/applied_mechanics_dynamics/radial-transverse-acceleration
- Whole chapter: https://www.bctengineer.com/md/semester_3/applied_mechanics_dynamics/chapter-1

## Question

Derive the radial and transverse components of acceleration when a particle moves in a curvilinear path.

## Short answer

Using polar coordinates with unit vectors e_r, e_θ, the time derivatives are e_ṙ = θ̇ e_θ and e_θ̇ = −θ̇ e_r. Differentiating r = r e_r twice yields a = (r̈ − rθ̇²) e_r + (rθ̈ + 2ṙθ̇) e_θ.

## Answer

A particle moving on a plane curve is located by polar coordinates $(r,\theta)$. Using the unit vectors $\hat e_r$ (along the radius vector) and $\hat e_\theta$ (perpendicular to it, in the direction of increasing $\theta$), velocity and acceleration are found by differentiating $\vec r = r\hat e_r$.

```text
        y
        |        P
        |       /|
        |    r / | e_theta
        |     /  |
        |    /   -> e_r along OP
        |   /\ theta
        +--------------- x
        O
```

#### Unit-vector derivatives
$$
\hat e_r=\cos\theta\,\hat i+\sin\theta\,\hat j,\qquad \hat e_\theta=-\sin\theta\,\hat i+\cos\theta\,\hat j
$$

Differentiating with respect to time:

$$
\dot{\hat e}_r=\dot\theta\,\hat e_\theta,\qquad \dot{\hat e}_\theta=-\dot\theta\,\hat e_r
$$

#### Velocity
$$
\vec v=\frac{d}{dt}(r\hat e_r)=\dot r\,\hat e_r+r\dot\theta\,\hat e_\theta
$$

So $v_r=\dot r$ and $v_\theta=r\dot\theta$.

#### Acceleration
$$
\begin{aligned}
\vec a=\frac{d\vec v}{dt}&=\ddot r\,\hat e_r+\dot r\,\dot{\hat e}_r+(\dot r\dot\theta+r\ddot\theta)\hat e_\theta+r\dot\theta\,\dot{\hat e}_\theta\\
&=\ddot r\,\hat e_r+\dot r\dot\theta\,\hat e_\theta+(\dot r\dot\theta+r\ddot\theta)\hat e_\theta-r\dot\theta^{2}\hat e_r
\end{aligned}
$$

Collecting terms:

$$
\vec a=(\ddot r-r\dot\theta^{2})\,\hat e_r+(r\ddot\theta+2\dot r\dot\theta)\,\hat e_\theta
$$

- Radial component: $a_r=\ddot r-r\dot\theta^{2}$
- Transverse component: $a_\theta=r\ddot\theta+2\dot r\dot\theta$
- Magnitude: $a=\sqrt{a_r^{2}+a_\theta^{2}}$

Here $-r\dot\theta^2$ is the centripetal term and $2\dot r\dot\theta$ is the Coriolis term. For motion on a circle ($r$ constant) $a_r=-r\dot\theta^2$ and $a_\theta=r\ddot\theta$.

## Check yourself

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

1. In polar coordinates, what are the time derivatives of the unit vectors?
   A. de_r/dt = theta_dot e_theta and de_theta/dt = -theta_dot e_r
   B. de_r/dt = theta_dot e_r and de_theta/dt = theta_dot e_theta
   C. de_r/dt = -theta_dot e_theta and de_theta/dt = theta_dot e_r
   D. de_r/dt = 0 and de_theta/dt = 0 because they have unit length
   Answer: A. Rotating e_r by dtheta tilts it towards e_theta, while e_theta tilts towards -e_r. (Key point: Unit-vector derivatives)

2. What are the radial and transverse components of velocity?
   A. v_r = r theta_dot, v_theta = r_dot
   B. v_r = r_dot, v_theta = r theta_dot
   C. v_r = r_dot, v_theta = r theta_double_dot
   D. v_r = r_double_dot, v_theta = r theta_dot
   Answer: B. Differentiating r e_r gives r_dot e_r + r theta_dot e_theta. (Key point: Velocity components)

3. Which expression is the radial component of acceleration?
   A. r_double_dot + r theta_dot^2
   B. r_double_dot - r theta_dot^2
   C. r theta_double_dot + 2 r_dot theta_dot
   D. r_double_dot - 2 r_dot theta_dot
   Answer: B. Collecting e_r terms gives a_r = r_double_dot - r theta_dot^2, with -r theta_dot^2 being the centripetal term. (Key point: Radial acceleration)

4. Which expression is the transverse component of acceleration, and which part is the Coriolis term?
   A. r theta_double_dot + 2 r_dot theta_dot; the Coriolis term is 2 r_dot theta_dot
   B. r theta_double_dot - 2 r_dot theta_dot; the Coriolis term is r theta_double_dot
   C. r theta_double_dot + r_dot theta_dot; the Coriolis term is r_dot theta_dot
   D. r theta_double_dot + 2 r_dot theta_dot; the Coriolis term is r theta_double_dot
   Answer: A. r_dot theta_dot appears twice (once from each differentiation), giving the 2 r_dot theta_dot Coriolis term. (Key point: Transverse acceleration and Coriolis term)

## Related in Applied Mechanics (Dynamics)

- [Radial and transverse velocity and acceleration](https://www.bctengineer.com/question.md?key=semester_3%2Fapplied_mechanics_dynamics%3Aradial-transverse-velocity-acceleration) — Applied Mechanics (Dynamics), chapter 1

## Related in other subjects

- [Radial and transverse components of acceleration](https://www.bctengineer.com/question.md?key=semester_3%2Fengineering_mechanics%3Apolar-components-velocity-acceleration) — Engineering Mechanics, chapter 2
- [Vibrating particle velocity, acceleration and path](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Avibrating-particle-x-100-sin-pi-t) — Applied Mechanics, chapter 8
- [Radial and transverse force on block B](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Ablock-b-polar-radial-transverse-force) — Applied Mechanics, chapter 9
- [Velocity and acceleration of particle on curve](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Avelocity-acceleration-particle-curve) — Mathematics, chapter 5, asked 2×