Study guide
3.1 Kinematics & Motion
NAPT Navy Advanced Programs Test 2026
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1 Introduction
About the NAPT & Navy Nuclear Field Program NFa & NFb Qualification Routes and Scoring How to Use This Guide
2 Chapter 1: Mathematics I — Arithmetic & Algebra
1.1 Arithmetic, Fractions, Decimals & Percentages 1.2 Ratios, Proportions & Unit Conversions 1.3 Algebra I: Linear Equations & Inequalities 1.4 Algebra II: Exponents, Radicals & Quadratic Equations
3 Chapter 2: Mathematics II — Geometry, Trigonometry & Word Problems
2.1 Geometry: Area, Volume & Right Triangles 2.2 Trigonometry Basics 2.3 Word Problems & Multi-Step Reasoning
4 Chapter 3: Physics I — Mechanics & Energy
3.1 Kinematics & Motion 3.2 Forces & Newton's Laws 3.3 Work, Energy & Power
5 Chapter 4: Physics II — Thermodynamics, Electricity & Waves
4.1 Thermodynamics & Heat Transfer 4.2 Electricity: Ohm's Law & Series/Parallel Circuits 4.3 Kirchhoff's Laws & Magnetism 4.4 Waves, Fluids & Pressure
6 Chapter 5: Chemistry I — Atomic Structure & Reactions
5.1 Atomic Structure & the Periodic Table 5.2 Chemical Bonding & Reactions 5.3 Stoichiometry & the Mole Concept
7 Chapter 6: Chemistry II — Gas Laws, Solutions & Thermochemistry
6.1 Gas Laws 6.2 Solutions, Acids & Bases 6.3 States of Matter & Thermochemistry
8 Chapter 7: Nuclear Science
7.1 Atomic & Nuclear Structure, Isotopes 7.2 Radioactivity, Decay & Half-Life 7.3 Nuclear Fission, Chain Reactions & Critical Mass 7.4 Reactor Basics & Radiation Safety
9 Chapter 8: Administration Boundaries & Editorial Review
8.1 Administration Boundaries & Personal Readiness 8.2 Integrated Review & Local Practice
3.1 Kinematics & Motion
Key Takeaways Explain This Topic AI
The four kinematics equations — v = v0 + at, Δx = v0t + ½at², v² = v0² + 2aΔx, and Δx = [(v0+v)/2]t — describe any constant-acceleration motion problem.
Distance and speed are scalars with no direction; displacement and velocity are vectors that include direction.
Free-fall problems use a = g = 9.8 m/s² (32.2 ft/s²) in the same kinematics equations as any other constant-acceleration problem.
Projectile motion splits into two independent components: constant horizontal velocity and constant vertical acceleration (g) downward.
Solving the vertical component for time of flight first, then applying that time horizontally, finds the range of any horizontally launched projectile.
Last updated: August 2026
Kinematics is included as transferable introductory physics, with standard relationships among displacement, velocity, acceleration, and time. The Navy has not publicly identified a NAPT physics section or the frequency of mechanics questions.
Distance, Displacement, Speed, and Velocity
Not every question asking how far or how fast means the same thing in physics, so distinguish the following quantities.
Distance is a scalar quantity — total path length traveled, with no direction attached. Walk 3 meters east and then 3 meters west, and you have traveled a distance of 6 meters.
Displacement is a vector quantity — the straight-line change in position, including direction. In that same example, your displacement is 0 meters because you ended up back where you started.
Speed is distance divided by time (a scalar). Velocity is displacement divided by time (a vector) — it carries both magnitude and direction.
Acceleration is the rate at which velocity changes over time, measured in meters per second squared (m/s²) in the metric system or feet per second squared (ft/s²) in the imperial system. An object speeding up, slowing down, or changing direction is all accelerating — deceleration is simply acceleration directed opposite to the motion.
- Quantity Type Common Symbol SI Unit
- Distance / Displacement scalar / vector d or Δx meter (m)
- Speed / Velocity scalar / vector v meters per second (m/s)
- Acceleration vector a meters per second squared (m/s²)
- Time scalar t second (s)
- The Four Kinematics Equations
For motion with constant acceleration , four standard equations relate position, velocity, acceleration, and time: Each equation omits one of the five key variables (v0, v, a, t, Δx), so picking the right equation just means picking the one that does not require the variable you were not given.
- Equation Missing Variable Use When You Know
- v = v0 + at Δx initial velocity, acceleration, time
- Δx = v0t + ½at² v initial velocity, acceleration, time
- v² = v0² + 2aΔx t initial velocity, acceleration, displacement
- Δx = [(v0 + v) / 2]t a both velocities and time
Where v0 = initial velocity, v = final velocity, a = acceleration, t = time, and Δx = displacement.
Worked Example 1: An Accelerating Boat
A small patrol boat starts from rest alongside the pier and accelerates at a constant 2.5 m/s² for 6 seconds before leveling off.
Find the final velocity. Since we know v0 (0 m/s), a (2.5 m/s²), and t (6 s), use v = v0 + at:
v = 0 + (2.5)(6) = 15 m/s
Find the distance covered. Use Δx = v0t + ½at²:
Δx = (0)(6) + ½(2.5)(6²) = 0 + ½(2.5)(36) = 45 meters
Because acceleration is constant, velocity grows in a straight line over time — the chart below plots the boat's velocity at each second of the run.
Worked Example 2: Free Fall
A wrench slips from a sailor's hand and falls from a height of 44.1 meters above the deck. Ignoring air resistance, gravity accelerates it downward at g = 9.8 m/s².
Find the time to hit the deck. Starting from rest, use Δx = v0t + ½at² with v0 = 0:
44.1 = ½(9.8)t² → t² = 44.1 / 4.9 = 9 → t = 3 seconds
Find the impact velocity. Use v = v0 + at:
v = 0 + (9.8)(3) = 29.4 m/s
Free-fall problems are just kinematics problems with a = g = 9.8 m/s² (or 32.2 ft/s² in imperial units) substituted in — nothing new to memorize.
Projectile Motion: Two Independent Problems in One
A projectile is any object moving under gravity alone after being launched, thrown, or dropped. A central introductory idea is that projectile motion can be separated into two independent motions:
Direction What Happens Governing Fact
Horizontal Constant velocity (zero acceleration) Nothing pushes or slows the object sideways (ignoring air resistance)
Vertical Constant acceleration, g = 9.8 m/s² downward Gravity acts the entire flight, independent of horizontal motion
Because the two directions never affect each other, you solve the vertical problem first to find time of flight , then plug that same time into the horizontal problem to find range (horizontal distance traveled).
Worked Example 3: Horizontal Launch
A signal flare is launched horizontally at 20 m/s from a platform 78.4 meters above the water.
Step 1 — Vertical: find time of flight. The flare has no initial vertical velocity, so use Δx = ½gt²:
78.4 = ½(9.8)t² → t² = 78.4 / 4.9 = 16 → t = 4 seconds
Step 2 — Horizontal: find range. Horizontal velocity never changes, so distance = velocity × time:
range = (20 m/s)(4 s) = 80 meters
Bonus — impact speed. The vertical velocity at splashdown is v = gt = (9.8)(4) = 39.2 m/s. Combining the horizontal (20 m/s) and vertical (39.2 m/s) components with the Pythagorean theorem gives a total impact speed of about 44 m/s — noticeably faster than either component alone, because velocity is a vector.
This same two-step method — solve the vertical component for time, then use that time horizontally — handles every horizontal-launch projectile problem you will see, whether it is a dropped tool, a thrown heaving line, or a launched flare.
Section Takeaways
Four kinematics equations cover any constant-acceleration motion; pick the one that omits the variable you were not given.
Distance and speed are scalars (no direction); displacement and velocity are vectors (direction matters).
Free fall is kinematics with a = g = 9.8 m/s² (32.2 ft/s²) substituted in.
Projectile motion splits into two independent problems: horizontal (constant velocity) and vertical (constant acceleration g).
Solve the vertical component for time of flight first, then use that time to find horizontal range.