Particle Curvilinear Motion

Dynamics · FE Reference Handbook section

Dynamics

7 formulas

10 exam-style examples

~59 min

All Dynamics lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Particle Curvilinear Motion within Dynamics. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what particle curvilinear motion describes physically and when it applies.
State every one of the 7 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: g = 32.2 ft/s² or 9.81 m/s²; never mix mass and weight.

Lecture

Why this section exists. Particle Curvilinear Motion is the part of Dynamics that lets you connect a particle or rigid body moving under known forces to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as kinematics first, then a work-energy or impulse-momentum shortcut. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. g = 32.2 ft/s² or 9.81 m/s²; never mix mass and weight. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Crane lowering a steel plate girder onto bridge bearings while ironworkers guide it. — Photo 1. Where this shows up in practice: particle curvilinear motion.
Capstone Studio instructional photograph

Dynamics — Particle Curvilinear Motion: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a particle or rigid body moving under known forces. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 7 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

vz	Quantity produced by "vz = zo az = zp" — read its definition and unit from the handbook line directly above the equation.
v	Quantity produced by "v=s" — read its definition and unit from the handbook line directly above the equation.
at	Quantity produced by "at = v = =v" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

x, y, z Coordinates r, i, z Coordinates
n, t, b Coordinates
dv dv
dt ds
v2 [1 + (dy/dx)2[3/2
d 2y

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Projectile range and peak height — Particle Curvilinear Motion

Material is thrown from a conveyor at 60 ft/s at 43° above horizontal onto ground at the same level. Find the flight time, range and maximum height.

Given

v₀ = 60 ft/s
θ = 43°
g = 32.2 ft/s²

Find

t_flight, R and H

Start with the thinking

Horizontal motion is constant velocity; vertical motion is constant acceleration.
Time links the two components.

Step-by-step solution

Components
$v_{x} = 60 cos 4 3^{\circ} = 43.88 f t / s, v_{γ} = 60 s in 4 3^{\circ} = 40.92 f t / s$
Flight time
$t = 2 v_{γ} / g$
Substituting
$t = 2 (40.92) /32.2 = 2.542 s$
Range — R = v₀ₓ·t = 43.88(2.542) = 111.5 ft
Peak height
$H = v_{γ}^{2} / (2 g) = 26.00 f t$

Answer: t ≈ 2.54 s, R ≈ 111.5 ft, H ≈ 26.0 ft

Why the other options are there

R = 152.5 ft (full speed used horizontally)
H = 52.0 ft (factor of 2 dropped)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 2

Normal and tangential components on a circular path — Particle Curvilinear Motion

A car travels a curve of radius 140.0 m at 8 m/s while speeding up at 1.5 m/s². Resolve the acceleration into normal and tangential components, find the total acceleration, and compute the angular velocity of the radius vector.

Given

ρ = 140.0 m
v = 8 m/s
a_t = 1.5 m/s²

Find

a_n, total a, and ω

Start with the thinking

The tangential component changes speed; the normal component changes direction and always points to the centre.
Total acceleration is the vector sum of the two perpendicular components.

Step-by-step solution

Formula
$a_{n} = v^{2} / ρ$
Substituting
$a_{n} = 8^{2} /140.0 = 0.457 m / s^{2}$
Formula
$a = (a_{t}^{2} + a_{n}^{2})$
Substituting
$a = (1. 5^{2} + 0.45 7^{2}) = 1.568 m / s^{2}$
Formula
$ω = v / ρ$
Substituting
$ω = 8/140.0 = 0.0571 r a d / s$

Answer: a_n = 0.46 m/s², total a = 1.57 m/s², ω = 0.057 rad/s

Why the other options are there

1.96 m/s² (components added directly)
8,960 m/s² (multiplied by radius)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 3

Projectile range and peak height — Particle Curvilinear Motion (2)

Material is thrown from a conveyor at 49 ft/s at 33° above horizontal onto ground at the same level. Find the flight time, range and maximum height.

Given

v₀ = 49 ft/s
θ = 33°
g = 32.2 ft/s²

Find

t_flight, R and H

Start with the thinking

Horizontal motion is constant velocity; vertical motion is constant acceleration.
Time links the two components.

Step-by-step solution

Components
$v_{x} = 49 cos 3 3^{\circ} = 41.09 f t / s, v_{γ} = 49 s in 3 3^{\circ} = 26.69 f t / s$
Flight time
$t = 2 v_{γ} / g$
Substituting
$t = 2 (26.69) /32.2 = 1.658 s$
Range — R = v₀ₓ·t = 41.09(1.658) = 68.1 ft
Peak height
$H = v_{γ}^{2} / (2 g) = 11.06 f t$

Answer: t ≈ 1.66 s, R ≈ 68 ft, H ≈ 11.1 ft

Why the other options are there

R = 81 ft (full speed used horizontally)
H = 22.1 ft (factor of 2 dropped)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 4

Normal and tangential components on a circular path — Particle Curvilinear Motion (2)

A car travels a curve of radius 140.0 m at 32 m/s while speeding up at 2.5 m/s². Resolve the acceleration into normal and tangential components, find the total acceleration, and compute the angular velocity of the radius vector.

Given

ρ = 140.0 m
v = 32 m/s
a_t = 2.5 m/s²

Find

a_n, total a, and ω

Start with the thinking

The tangential component changes speed; the normal component changes direction and always points to the centre.
Total acceleration is the vector sum of the two perpendicular components.

Step-by-step solution

Formula
$a_{n} = v^{2} / ρ$
Substituting
$a_{n} = 3 2^{2} /140.0 = 7.314 m / s^{2}$
Formula
$a = (a_{t}^{2} + a_{n}^{2})$
Substituting
$a = (2. 5^{2} + 7.31 4^{2}) = 7.730 m / s^{2}$
Formula
$ω = v / ρ$
Substituting
$ω = 32/140.0 = 0.2286 r a d / s$

Answer: a_n = 7.31 m/s², total a = 7.73 m/s², ω = 0.229 rad/s

Why the other options are there

9.81 m/s² (components added directly)
143,360 m/s² (multiplied by radius)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 5

Projectile range and peak height — Particle Curvilinear Motion (3)

Material is thrown from a conveyor at 86 ft/s at 49° above horizontal onto ground at the same level. Find the flight time, range and maximum height.

Given

v₀ = 86 ft/s
θ = 49°
g = 32.2 ft/s²

Find

t_flight, R and H

Start with the thinking

Horizontal motion is constant velocity; vertical motion is constant acceleration.
Time links the two components.

Step-by-step solution

Components
$v_{x} = 86 cos 4 9^{\circ} = 56.42 f t / s, v_{γ} = 86 s in 4 9^{\circ} = 64.91 f t / s$
Flight time
$t = 2 v_{γ} / g$
Substituting
$t = 2 (64.91) /32.2 = 4.031 s$
Range — R = v₀ₓ·t = 56.42(4.031) = 227.5 ft
Peak height
$H = v_{γ}^{2} / (2 g) = 65.41 f t$

Answer: t ≈ 4.03 s, R ≈ 227.5 ft, H ≈ 65.4 ft

Why the other options are there

R = 346.7 ft (full speed used horizontally)
H = 130.8 ft (factor of 2 dropped)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 6

Normal and tangential components on a circular path — Particle Curvilinear Motion (3)

A car travels a curve of radius 125.0 m at 12 m/s while speeding up at 2.0 m/s². Resolve the acceleration into normal and tangential components, find the total acceleration, and compute the angular velocity of the radius vector.

Given

ρ = 125.0 m
v = 12 m/s
a_t = 2.0 m/s²

Find

a_n, total a, and ω

Start with the thinking

The tangential component changes speed; the normal component changes direction and always points to the centre.
Total acceleration is the vector sum of the two perpendicular components.

Step-by-step solution

Formula
$a_{n} = v^{2} / ρ$
Substituting
$a_{n} = 1 2^{2} /125.0 = 1.152 m / s^{2}$
Formula
$a = (a_{t}^{2} + a_{n}^{2})$
Substituting
$a = (2. 0^{2} + 1.15 2^{2}) = 2.308 m / s^{2}$
Formula
$ω = v / ρ$
Substituting
$ω = 12/125.0 = 0.0960 r a d / s$

Answer: a_n = 1.15 m/s², total a = 2.31 m/s², ω = 0.096 rad/s

Why the other options are there

3.15 m/s² (components added directly)
18,000 m/s² (multiplied by radius)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 7

Projectile range and peak height — Particle Curvilinear Motion (4)

Material is thrown from a conveyor at 101 ft/s at 43° above horizontal onto ground at the same level. Find the flight time, range and maximum height.

Given

v₀ = 101 ft/s
θ = 43°
g = 32.2 ft/s²

Find

t_flight, R and H

Start with the thinking

Horizontal motion is constant velocity; vertical motion is constant acceleration.
Time links the two components.

Step-by-step solution

Components
$v_{x} = 101 cos 4 3^{\circ} = 73.87 f t / s, v_{γ} = 101 s in 4 3^{\circ} = 68.88 f t / s$
Flight time
$t = 2 v_{γ} / g$
Substituting
$t = 2 (68.88) /32.2 = 4.278 s$
Range — R = v₀ₓ·t = 73.87(4.278) = 316.0 ft
Peak height
$H = v_{γ}^{2} / (2 g) = 73.68 f t$

Answer: t ≈ 4.28 s, R ≈ 316.0 ft, H ≈ 73.7 ft

Why the other options are there

R = 432.1 ft (full speed used horizontally)
H = 147.4 ft (factor of 2 dropped)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 8

Normal and tangential components on a circular path — Particle Curvilinear Motion (4)

A car travels a curve of radius 115.0 m at 21 m/s while speeding up at 1.5 m/s². Resolve the acceleration into normal and tangential components, find the total acceleration, and compute the angular velocity of the radius vector.

Given

ρ = 115.0 m
v = 21 m/s
a_t = 1.5 m/s²

Find

a_n, total a, and ω

Start with the thinking

The tangential component changes speed; the normal component changes direction and always points to the centre.
Total acceleration is the vector sum of the two perpendicular components.

Step-by-step solution

Formula
$a_{n} = v^{2} / ρ$
Substituting
$a_{n} = 2 1^{2} /115.0 = 3.835 m / s^{2}$
Formula
$a = (a_{t}^{2} + a_{n}^{2})$
Substituting
$a = (1. 5^{2} + 3.83 5^{2}) = 4.118 m / s^{2}$
Formula
$ω = v / ρ$
Substituting
$ω = 21/115.0 = 0.1826 r a d / s$

Answer: a_n = 3.83 m/s², total a = 4.12 m/s², ω = 0.183 rad/s

Why the other options are there

5.33 m/s² (components added directly)
50,715 m/s² (multiplied by radius)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 9

Projectile range and peak height — Particle Curvilinear Motion (5)

Material is thrown from a conveyor at 118 ft/s at 57° above horizontal onto ground at the same level. Find the flight time, range and maximum height.

Given

v₀ = 118 ft/s
θ = 57°
g = 32.2 ft/s²

Find

t_flight, R and H

Start with the thinking

Horizontal motion is constant velocity; vertical motion is constant acceleration.
Time links the two components.

Step-by-step solution

Components
$v_{x} = 118 cos 5 7^{\circ} = 64.27 f t / s, v_{γ} = 118 s in 5 7^{\circ} = 98.96 f t / s$
Flight time
$t = 2 v_{γ} / g$
Substituting
$t = 2 (98.96) /32.2 = 6.147 s$
Range — R = v₀ₓ·t = 64.27(6.147) = 395.0 ft
Peak height
$H = v_{γ}^{2} / (2 g) = 152.1 f t$

Answer: t ≈ 6.15 s, R ≈ 395.0 ft, H ≈ 152.1 ft

Why the other options are there

R = 725.3 ft (full speed used horizontally)
H = 304.2 ft (factor of 2 dropped)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Example 10

Normal and tangential components on a circular path — Particle Curvilinear Motion (5)

A car travels a curve of radius 85 m at 34 m/s while speeding up at 1.0 m/s². Resolve the acceleration into normal and tangential components, find the total acceleration, and compute the angular velocity of the radius vector.

Given

ρ = 85 m
v = 34 m/s
a_t = 1.0 m/s²

Find

a_n, total a, and ω

Start with the thinking

The tangential component changes speed; the normal component changes direction and always points to the centre.
Total acceleration is the vector sum of the two perpendicular components.

Step-by-step solution

Formula
$a_{n} = v^{2} / ρ$
Substituting
$a_{n} = 3 4^{2} /85 = 13.600 m / s^{2}$
Formula
$a = (a_{t}^{2} + a_{n}^{2})$
Substituting
$a = (1. 0^{2} + 13.60 0^{2}) = 13.637 m / s^{2}$
Formula
$ω = v / ρ$
Substituting
$ω = 34/85 = 0.4000 r a d / s$

Answer: a_n = 13.60 m/s², total a = 13.64 m/s², ω = 0.400 rad/s

Why the other options are there

14.60 m/s² (components added directly)
98,260 m/s² (multiplied by radius)

Reference: FE Reference Handbook — Dynamics → Particle Curvilinear Motion

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a particle or rigid body moving under known forces, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Particle Curvilinear Motion contains 7 relations; you must be able to find this page in under 15 seconds.
Exam style: kinematics first, then a work-energy or impulse-momentum shortcut.
Unit rule: g = 32.2 ft/s² or 9.81 m/s²; never mix mass and weight.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

g = 32.2 ft/s² or 9.81 m/s²; never mix mass and weight
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.