Kinematics of Particles and Rigid Bodies

NCEES FE Civil spec: Dynamics, A. Kinematics (e.g., particles, rigid bodies). FE Reference Handbook: Dynamics chapter, kinematics.

Kinematics describes motion (position, velocity and acceleration) without asking what causes it. On the FE Civil exam the questions are short: a vehicle braking at a constant rate, a projectile, a car on a curve, or a wheel spinning up. If you can pick the right constant-acceleration equation quickly, most of these take under two minutes.

What the NCEES spec covers

Section A covers kinematics of particles and rigid bodies: straight-line motion, projectile motion, curvilinear motion with normal and tangential components, and rotation about a fixed axis. Relative motion and instant centers can appear but are less common for civil candidates.

Key concepts

Definitions. Velocity is the time derivative of position, and acceleration is the derivative of velocity. If acceleration varies, integrate. If it is constant, use the closed-form equations.
Projectiles. Horizontal and vertical motions are independent. Horizontal velocity is constant; vertical acceleration is g downward.
Curvilinear motion. Tangential acceleration changes speed; normal (centripetal) acceleration changes direction and points toward the center of curvature.
Rotation. Angular equations mirror the linear ones, with θ, ω and α replacing s, v and a. A point at radius r has speed rω.

Core formulas

Constant linear acceleration:

v = v_{0} + a t s = s_{0} + v_{0} t + \frac{1}{2} a t^{2} v^{2} = v_{0}^{2} + 2 a (s - s_{0})

Constant angular acceleration:

ω = ω_{0} + α t θ = θ_{0} + ω_{0} t + \frac{1}{2} α t^{2} ω^{2} = ω_{0}^{2} + 2 α (θ - θ_{0})

Circular motion and rigid body rotation:

v = r ω a_{t} = r α a_{n} = \frac{v ^{2}}{r} = r ω^{2}

Projectile launched from level ground at speed $v_{0}$ and angle θ:

t_{flight} = \frac{2 v _{0} sin θ}{g} R = \frac{v _{0}^{2} sin 2 θ}{g} h_{m a x} = \frac{( v _{0} sin θ ) ^{2}}{2 g}

Where it is in the FE Reference Handbook

The Dynamics chapter opens with kinematics: particle motion in rectangular, normal-tangential and polar coordinates, constant acceleration, projectile motion and rigid body rotation. The range and flight-time shortcuts above are derived from those, so practice the derivation in case the handbook form is different.

Worked example

A ball is kicked from level ground at 25 m/s, 40° above the horizontal. Find the time of flight, horizontal range and maximum height. Neglect air resistance.

Step 1: components.

v_{x} = 25 cos 4 0^{\circ} = 19.15 m/s v_{y} = 25 sin 4 0^{\circ} = 16.07 m/s

Step 2: time of flight (vertical displacement returns to zero):

t = \frac{2 ( 16.07 )}{9.81} = 3.28 s

Step 3: range and maximum height.

R = v_{x} t = 19.15 (3.276) = 62.7 m h_{m a x} = \frac{16.0 7 ^{2}}{2 ( 9.81 )} = 13.2 m

Check with the range shortcut: $2 5^{2} sin 8 0^{\circ} /9.81 = 62.7 m$ . ✓

Common mistakes

Using constant-acceleration equations when acceleration depends on time or position.
Giving g the wrong sign. Choose up or down as positive and stay consistent.
Using the range formula when launch and landing heights differ. Solve the vertical quadratic instead.
Mixing rpm with rad/s. Multiply rpm by 2π/60.
Forgetting the centripetal component when asked for total acceleration on a curve.

See the constant acceleration example for a straight-line problem.

Practice Problems

Try these from the free PolyCalc FE Civil problem bank. Each one has a full worked solution.

An object has initial velocity 10 m/s and constant acceleration 2 m/s² in the same direction for 5 s. What is its final velocity? See the worked solution or practice the Dynamics & Seismic deck
An object starts from rest and accelerates in a straight line at a constant 3 m/s² for 4 s. What distance does it travel? See the worked solution or practice the Dynamics & Seismic deck
A vehicle travels at 8 m/s around a circular path of radius 20 m. What is its centripetal acceleration? See the worked solution or practice the Dynamics & Seismic deck
A ball is launched from level ground at 20 m/s, 30° above the horizontal. Neglecting air resistance, its horizontal range is most nearly: See the worked solution or practice the Dynamics & Seismic deck

Related Calculators and Examples

Frequently Asked Questions

When can I use the constant acceleration equations?

Only when acceleration is constant, such as free fall or uniform braking. If acceleration is given as a function of time, velocity or position, integrate the definitions instead.

What is the difference between tangential and normal acceleration?

Tangential acceleration changes the speed and acts along the path. Normal acceleration equals v^2 / r, points toward the center of curvature and changes only the direction of motion.

How do I convert rpm to rad/s?

Multiply by 2 pi and divide by 60. For example, 1,200 rpm is 1,200 x 2 pi / 60 = 125.7 rad/s.

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