Dynamics

9% of exam

Kinematics, Newton's second law, work-energy and impulse-momentum, rotation, mass moment of inertia, and natural frequency.

5 concepts

Exam areas

A. Particle and rigid-body kinematics

Kinematics of Particles and Rigid Bodies

Position, velocity, and acceleration in rectilinear, projectile, normal-tangential, and polar form, plus relative motion and rigid-body rotation with instant centers.

Kinematics is the geometry of motion: where a body is, how fast it moves, and how that velocity changes — with no reference to forces or mass. It is the foundation every other dynamics topic stands on, and on the FE it is where the easiest points and the silliest mistakes both live. The Dynamics chapter of the FE Reference Handbook opens here, and the single skill it rewards is choosing the right coordinate description — rectilinear, projectile, normal-tangential, or polar — for the motion in front of you. Pick the wrong one and a two-line problem becomes a page of trigonometry; pick the right one and the answer falls out.

Velocity and acceleration are derivatives

Everything starts from one idea: velocity is the time derivative of position and acceleration is the time derivative of velocity. For one-dimensional motion along a coordinate

s

v=\dot s

and

a=\dot v=\ddot s

. A fourth relation, obtained by the chain rule

a=v\,dv/ds

, is the one to reach for whenever a problem links velocity to position but never mentions time — it lets you integrate

a\,ds=v\,dv

directly.

v=\dfrac{ds}{dt},\qquad a=\dfrac{dv}{dt}=v\,\dfrac{dv}{ds}

Constant acceleration and free fall

When acceleration is constant the integrals collapse to three workhorse formulas. These cover free fall (

a=-g

), braking, and any uniformly accelerated motion. The time-free form

v^2=v_0^2+2a(s-s_0)

is the fastest route whenever the problem gives you a distance but asks nothing about the time taken. Use these only when

a

is genuinely constant — a spring or drag force makes

a

vary, and these equations no longer apply.

v=v_0+a\,t,\qquad s=s_0+v_0 t+\tfrac{1}{2}a t^2,\qquad v^2=v_0^2+2a\,(s-s_0)

Projectile motion: two independent axes

Projectile motion is just constant acceleration applied separately to two axes. The horizontal axis has zero acceleration, so

v_x

stays constant; the vertical axis carries the full gravity,

a_y=-g

. The two are linked only through the shared time

t

. Resolve the launch speed once,

v_{x}=v_0\cos\theta

and

v_{y0}=v_0\sin\theta

, then treat each axis with its own constant-acceleration equations. Maximum height occurs where

v_y=0

x=v_0\cos\theta\,t,\qquad y=v_0\sin\theta\,t-\tfrac{1}{2}g t^2

Normal and tangential components

When a particle follows a known curved path, the natural axes are tangential (along the path) and normal (toward the center of curvature). The tangential acceleration changes the speed; the normal acceleration changes the direction and always points inward. Even at constant speed a curving particle accelerates, because

a_n=v^2/\rho

is nonzero whenever the path bends. The total acceleration magnitude is the vector sum

a=\sqrt{a_t^2+a_n^2}

, with

\rho

the radius of curvature.

a_t=\dfrac{dv}{dt},\qquad a_n=\dfrac{v^2}{\rho},\qquad a=\sqrt{a_t^2+a_n^2}

Polar (radial-transverse) components

For motion described by a radius

r

and angle

\theta

from a fixed point — a radar track, a slotted arm, a rotating mechanism — polar components are cleanest. The acceleration carries two extra terms beyond the obvious

\ddot r

and

r\ddot\theta

: the centripetal term

-r\dot\theta^2

and the Coriolis term

2\dot r\dot\theta

. Forgetting these two terms is the classic polar-coordinate error.

v_r=\dot r,\quad v_\theta=r\dot\theta,\qquad a_r=\ddot r-r\dot\theta^2,\quad a_\theta=r\ddot\theta+2\dot r\dot\theta

Relative motion of two particles

When two particles move at once, write each absolute quantity as a base value plus a relative one:

\mathbf{v}_A=\mathbf{v}_B+\mathbf{v}_{A/B}

. These are vector equations — resolve into

x

and

y

before adding. Relative motion is how the FE asks about two cars at an intersection, a swimmer in a current, or a package dropped from a moving plane. Read

\mathbf{v}_{A/B}

as 'velocity of

A

as seen by an observer riding on

B

\mathbf{r}_A=\mathbf{r}_B+\mathbf{r}_{A/B},\qquad \mathbf{v}_A=\mathbf{v}_B+\mathbf{v}_{A/B},\qquad \mathbf{a}_A=\mathbf{a}_B+\mathbf{a}_{A/B}

Rigid-body rotation and the instant center

A rigid body rotating about a fixed axis behaves exactly like rectilinear motion with angle replacing distance:

\omega=\dot\theta

\alpha=\dot\omega

, and constant-

\alpha

formulas mirror the linear ones. Every point on the body moves with

v=r\omega

and has

a_t=r\alpha

a_n=r\omega^2

. For general plane motion (translating and rotating at once, like a wheel that rolls), there is at every instant one point with zero velocity — the instantaneous center of rotation. The whole body momentarily rotates about it, so

v=\omega\,d

where

d

is the distance from a point to the instant center. For a wheel rolling without slipping, the instant center is the contact point: the top of the wheel moves at

2\omega r

and the center at

\omega r

v=r\omega,\qquad a_t=r\alpha,\qquad a_n=r\omega^2,\qquad v_P=\omega\,d_{P/IC}

Exam strategy

First classify the motion and pick coordinates: straight line and constant force → rectilinear constant-

a

; launched object → projectile (split the axes); known curved path → normal-tangential; radius-and-angle description → polar; rolling or linked bodies → rigid-body with the instant center. Resolve angles once at the start and carry components, not magnitudes. Keep a clear sign convention (up and right positive is the safe default) and watch units — the FE mixes SI and USCS, so confirm

g=9.81\ \text{m/s}^2

32.2\ \text{ft/s}^2

before substituting. When time is absent, reach for

v^2=v_0^2+2a\,\Delta s

a=v\,dv/ds

instead of grinding through

t

Key equations

Kinematic derivatives

v=\dfrac{ds}{dt},\qquad a=\dfrac{dv}{dt}=v\,\dfrac{dv}{ds}

Definitions of velocity and acceleration. The $v\,dv/ds$ form integrates motion that depends on position with no time given. $s$ in m (or ft), $v$ in m/s, $a$ in m/s².

Constant-acceleration set

v=v_0+a t,\quad s=s_0+v_0 t+\tfrac{1}{2}a t^2,\quad v^2=v_0^2+2a(s-s_0)

Use only when $a$ is constant (free fall, braking). The last form eliminates time.

Projectile equations

x=v_0\cos\theta\,t,\qquad y=v_0\sin\theta\,t-\tfrac{1}{2}g t^2

Constant acceleration on independent axes: $a_x=0$ , $a_y=-g$ . $\theta$ measured from horizontal.

Max height and range (level ground)

H=\dfrac{(v_0\sin\theta)^2}{2g},\qquad R=\dfrac{v_0^2\sin 2\theta}{g}

Apex height and horizontal range for launch and landing at the same elevation. Range is maximized at $\theta=45^\circ$ .

Normal-tangential acceleration

a_t=\dfrac{dv}{dt},\qquad a_n=\dfrac{v^2}{\rho},\qquad a=\sqrt{a_t^2+a_n^2}

$a_t$ changes speed, $a_n$ changes direction (points to center of curvature). $\rho$ = radius of curvature in m.

Polar velocity and acceleration

v_r=\dot r,\quad v_\theta=r\dot\theta,\qquad a_r=\ddot r-r\dot\theta^2,\quad a_\theta=r\ddot\theta+2\dot r\dot\theta

Radial and transverse velocity and acceleration. $-r\dot\theta^2$ is centripetal; $2\dot r\dot\theta$ is Coriolis. $\dot\theta=\omega$ in rad/s.

Relative motion

\mathbf{v}_A=\mathbf{v}_B+\mathbf{v}_{A/B},\qquad \mathbf{a}_A=\mathbf{a}_B+\mathbf{a}_{A/B}

Vector equations; resolve into components before adding. $\mathbf{v}_{A/B}$ = velocity of $A$ relative to $B$ .

Rotation kinematics (constant α)

\omega=\omega_0+\alpha t,\quad \theta=\theta_0+\omega_0 t+\tfrac{1}{2}\alpha t^2,\quad \omega^2=\omega_0^2+2\alpha(\theta-\theta_0)

Angular analogs of constant-acceleration motion. $\omega$ in rad/s, $\alpha$ in rad/s², $\theta$ in rad.

Point velocity on a rotating body

v=r\omega,\qquad a_t=r\alpha,\qquad a_n=r\omega^2

Speed and acceleration components of a point a distance $r$ from the rotation axis. $r$ in m.

Instantaneous center relation

v_P=\omega\,d_{P/IC}

Any point's speed equals $\omega$ times its distance to the instant center. For a wheel rolling without slip, IC is the contact point.

Worked examples

Braking distance from constant deceleration

Problem. A car travels at $30\ \text{m/s}$ when the driver brakes, producing a constant deceleration of $4\ \text{m/s}^2$ . How far does the car travel before stopping, and how long does it take?

Solution. Take the direction of travel as positive, so

a=-4\ \text{m/s}^2

v_0=30\ \text{m/s}

, final

v=0

. Distance from the time-free equation:

v^2=v_0^2+2a\,s \Rightarrow 0=30^2+2(-4)s

, so

s=\dfrac{900}{8}=112.5\ \text{m}

. Time from

v=v_0+at

0=30+(-4)t \Rightarrow t=7.50\ \text{s}

. Sanity check: average speed

15\ \text{m/s}\times 7.5\ \text{s}=112.5\ \text{m}

, consistent. Answers:

s=113\ \text{m}

t=7.50\ \text{s}

s=\dfrac{v_0^2}{2|a|}=\dfrac{30^2}{2(4)}=112.5\ \text{m}

Projectile range and maximum height

Problem. A ball is launched from ground level at $25\ \text{m/s}$ at $40^\circ$ above the horizontal. Find the maximum height, the time of flight, and the horizontal range. Use $g=9.81\ \text{m/s}^2$ .

Solution. Resolve:

v_x=25\cos 40^\circ=19.15\ \text{m/s}

v_{y0}=25\sin 40^\circ=16.07\ \text{m/s}

. Max height (where

v_y=0

H=\dfrac{v_{y0}^2}{2g}=\dfrac{16.07^2}{2(9.81)}=13.2\ \text{m}

. Time of flight (return to

y=0

T=\dfrac{2v_{y0}}{g}=\dfrac{2(16.07)}{9.81}=3.28\ \text{s}

. Range:

R=v_x T=19.15(3.276)=62.7\ \text{m}

. Sanity check:

R=\dfrac{v_0^2\sin 2\theta}{g}=\dfrac{25^2\sin 80^\circ}{9.81}=62.7\ \text{m}

, matches. Answers:

H=13.2\ \text{m}

T=3.28\ \text{s}

R=62.7\ \text{m}

Total acceleration on a curve

Problem. A car rounds a curve of radius $200\ \text{m}$ at $25\ \text{m/s}$ while its speed is increasing at $2\ \text{m/s}^2$ . Find the magnitude of its total acceleration.

Solution. Tangential component is given:

a_t=2\ \text{m/s}^2

. Normal component:

a_n=\dfrac{v^2}{\rho}=\dfrac{25^2}{200}=3.125\ \text{m/s}^2

. Total:

a=\sqrt{a_t^2+a_n^2}=\sqrt{2^2+3.125^2}=\sqrt{13.766}=3.71\ \text{m/s}^2

. Sanity check: the normal term dominates as expected for a fast turn, and the total exceeds each component but is less than their sum (

5.125

). Answer:

a=3.71\ \text{m/s}^2

a=\sqrt{a_t^2+\left(\dfrac{v^2}{\rho}\right)^2}=\sqrt{2^2+3.125^2}=3.71\ \text{m/s}^2

Rolling wheel and the instant center

Problem. A wheel of radius $0.35\ \text{m}$ rolls without slipping at an angular velocity of $12\ \text{rad/s}$ . Find the velocity of the center and the velocity of the topmost point of the wheel.

Solution. The contact point is the instantaneous center of rotation (zero velocity), so

v=\omega d

measured from there. Center is

0.35\ \text{m}

above the contact point:

v_c=\omega r=12(0.35)=4.20\ \text{m/s}

. Top point is

2r=0.70\ \text{m}

above the contact point:

v_{top}=\omega(2r)=12(0.70)=8.40\ \text{m/s}

. Sanity check: the top moves at exactly twice the center speed, the signature of rolling without slip. Answers:

v_c=4.20\ \text{m/s}

v_{top}=8.40\ \text{m/s}

Common pitfalls

•Using a constant-acceleration equation when acceleration is not constant. A spring, drag, or position-dependent force makes $a$ vary — switch to $a=v\,dv/ds$ and integrate.
•Forgetting that a particle on a curve accelerates even at constant speed: $a_n=v^2/\rho$ is nonzero whenever the path bends. Reporting $a=0$ for steady cornering is wrong.
•Dropping the centripetal $-r\dot\theta^2$ or Coriolis $2\dot r\dot\theta$ terms in polar acceleration. Both appear whenever $r$ and $\theta$ change together.
•Mixing degrees and radians: $\omega$ , $\alpha$ , and $v=r\omega$ require radians. Convert rev/min to rad/s with $\times 2\pi/60$ before substituting.
•Treating relative-motion equations as scalar. $\mathbf{v}_A=\mathbf{v}_B+\mathbf{v}_{A/B}$ is a vector sum — resolve into components first, never just add magnitudes.
•Using $v=r\omega$ with $r$ measured from the wrong point. For general plane motion, measure $r$ from the instantaneous center, not the geometric center.
•Plugging in the wrong $g$ . Confirm $9.81\ \text{m/s}^2$ (SI) versus $32.2\ \text{ft/s}^2$ (USCS) — the FE uses both and mismatched units silently corrupt the answer.

References

NCEES FE Reference Handbook — Dynamics (Particle Kinematics, Kinematics of a Rigid Body)
Hibbeler, Engineering Mechanics: Dynamics — Chapters 12 and 16 — particle and rigid-body kinematics

B. Linear motion

Newton's Second Law and Kinetics

Turn forces into motion with F=ma: free-body equations, kinetic friction, normal force on inclines, connected systems, and the D'Alembert inertial-force method.

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C. Angular motion

Angular Motion and Mass Moment of Inertia

Rotational kinetics with ΣM=Iα: torque, mass moment of inertia for standard shapes, the parallel-axis theorem, radius of gyration, and rolling without slipping.

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F. Work, energy, and power

Work-Energy and Impulse-Momentum

The two great shortcuts of dynamics: work-energy (force over distance) and impulse-momentum (force over time), plus power, conservation laws, and impact with restitution.

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H. Vibrations

Free Vibration and Natural Frequency

Simple harmonic motion and the undamped natural frequency ωn=√(k/m): period, frequency, simple and compound pendulums, equivalent stiffness, and a damping primer.

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