Units, Vectors & Translational Motion

On the MCAT, physics never stands alone — it is the mathematical backbone underneath biology passages about blood flow, muscle contraction, joint loading, and nerve conduction. Content Category 4A opens with the basic vocabulary of motion: units, vectors, and the definitions of speed, velocity, and acceleration. Master this vocabulary first, because every later topic in mechanics (force, work, energy, equilibrium, and periodic motion) builds directly on top of it.

Units and Dimensions

The MCAT reports every quantity in the International System of Units (SI), the standard system of measurement built around base units for length, mass, and time.

Most mechanics quantities are derived units — combinations of the base units above. Force is measured in newtons (N), where 1 N = 1 kg·m/s². Energy and work are measured in joules (J), where 1 J = 1 kg·m²/s² = 1 N·m. Power is measured in watts (W), where 1 W = 1 J/s. Recognizing these unit breakdowns lets you check an answer choice by dimensional analysis: if a proposed formula for force produces units of kg·m²/s² (joules), something is wrong before you even think about the physics.

Since the MCAT allows no calculator, unit conversions must rely on clean mental-math shortcuts. The most useful one in mechanics: to convert kilometers per hour to meters per second, divide by 3.6 (or equivalently multiply by 5/18). A car traveling at 72 km/h is moving at 72 ÷ 3.6 = 20 m/s — a conversion you should be able to do in your head in a few seconds. Another frequent conversion is centimeters to meters (divide by 100) when a passage gives vessel diameters or joint displacements in cm.

Vectors and Vector Components

A scalar is a quantity with magnitude only (mass, distance, speed, energy, work). A vector is a quantity with both magnitude and direction (displacement, velocity, acceleration, force, momentum). The MCAT tests vectors constantly, because forces acting on a body — gravity pulling down, a muscle pulling at an angle, blood pressure pushing outward on a vessel wall — rarely point in convenient directions.

Any vector can be broken into perpendicular components, almost always horizontal (x) and vertical (y), using right-triangle relationships:

• Vx = V cos θ
• Vy = V sin θ

where θ is the angle measured from the horizontal axis. The MCAT favors triangles with easy trig values so no calculator is ever needed: 30-60-90 triangles (sin 30° = 0.5, cos 30° ≈ 0.87, cos 60° = 0.5, sin 60° ≈ 0.87) and the 3-4-5 triangle, which appears as a 37°-53°-90° triangle (sin 37° ≈ 0.6, cos 37° ≈ 0.8).

Worked example: A force of 10 N is applied at 37° above the horizontal. Using cos 37° ≈ 0.8 and sin 37° ≈ 0.6:

• Horizontal component: Fx = 10 N × 0.8 = 8 N
• Vertical component: Fy = 10 N × 0.6 = 6 N

Notice that 8 and 6 are legs of a 3-4-5 triangle scaled by 2 (6-8-10) — the Pythagorean check (6² + 8² = 36 + 64 = 100 = 10²) confirms the components are correct.

Vector addition works by adding components separately, then recombining: sum all the x-components, sum all the y-components, and apply the Pythagorean theorem to find the resultant magnitude.

Worked example: A person walks 3 km east, then 4 km north. What is their total displacement?

• x-components sum to 3 km; y-components sum to 4 km
• Resultant magnitude: √(3² + 4²) = √25 = 5 km
• Direction: arctan(4/3) ≈ 53° north of east

The straight-line displacement (5 km) is shorter than the total path walked (3 km + 4 km = 7 km) — displacement measures start-to-finish position change in a straight line, while distance traveled measures the total length of the path. In gait analysis, step length and stride length track distance along the path of the feet, while net displacement over a closed walking loop around a clinic hallway can be near zero even after thousands of meters walked.

Speed vs. Velocity

Speed is a scalar: total distance traveled divided by time elapsed. Velocity is a vector: displacement (the straight-line change in position, including direction) divided by time elapsed. These two quantities are equal only when an object moves in a straight line without ever reversing direction.

Average speed = total distance / total time Average velocity = total displacement / total time

Worked example: A runner completes one lap of a 400 m track in 50 seconds, finishing at the same point where they started.

• Average speed = 400 m / 50 s = 8 m/s
• Average velocity = 0 m / 50 s = 0 m/s (displacement is zero because start and finish coincide)

This is a favorite MCAT trap: a passage describes a fast-moving object (high speed) that nonetheless has zero or very low average velocity because it ends up back near its starting point.

Instantaneous velocity is the velocity at one specific moment — mathematically, the slope of a position-versus-time graph at a single point. Instantaneous speed is simply the magnitude of that instantaneous velocity vector. A car's speedometer displays instantaneous speed, not average speed.

In physiology, this distinction matters for blood flow: the average velocity of blood through a vessel over a full cardiac cycle differs from the instantaneous velocity during systole (peak ejection) versus diastole, because pulsatile flow speeds up and slows down with every heartbeat. A passage may report peak systolic velocity in the aorta (high instantaneous speed) while asking for mean velocity over the cycle — do not treat those as the same number.

Acceleration

Acceleration is the rate of change of velocity with respect to time:

a = Δv / Δt

Acceleration is a vector with SI units of m/s². An object accelerates whenever its speed changes, its direction changes, or both — so an object moving in a circle at constant speed is still accelerating, since its direction is constantly changing. That idea resurfaces later in equilibrium and circular-motion contexts and again when analyzing orbital and molecular motion.

For situations of constant acceleration, three kinematics equations connect position, velocity, acceleration, and time:

• v = v₀ + at
• Δx = v₀t + ½at²
• v² = v₀² + 2aΔx

The most common constant-acceleration scenario on the MCAT is free fall near Earth's surface, where the acceleration due to gravity is g = 9.8 m/s². For mental math, round to g ≈ 10 m/s² — the small error this introduces almost never changes which answer choice is correct, and the arithmetic becomes dramatically easier.

Worked example: A ball is dropped from rest and falls freely for 3 seconds. Using g ≈ 10 m/s² and v = v₀ + at:

v = 0 + (10 m/s²)(3 s) = 30 m/s

Worked example (distance fallen): How far does the same ball fall in those 3 seconds? Using Δx = v₀t + ½at²:

Δx = 0 + ½(10 m/s²)(3 s)² = ½(10)(9) = 45 m

Notice that doubling the fall time does not double the distance fallen — because distance depends on t², a 3-second fall covers 45 m, but a 6-second fall covers ½(10)(36) = 180 m, four times as far. This t² relationship is one of the most frequently tested numerical traps in MCAT kinematics passages.

MCAT trap list for kinematics:

• Confusing distance with displacement (lap around a track)
• Using average velocity when the question asks for average speed
• Forgetting that free-fall speed scales with t while distance scales with t²
• Treating g as a force instead of an acceleration (force is mg, not g alone)
• Assuming zero acceleration means zero velocity — zero acceleration means constant velocity, which can be nonzero