Force and Motion

Force and Motion is one of the largest topics in MCAT physics. The AAMC MCAT content outline expects you to master kinematics (displacement, velocity, acceleration), Newton's three laws, momentum and impulse, elastic vs inelastic collisions, and the equations of projectile motion. It is a high-yield Chem/Phys area.

AAMC content categories. This chapter spans kinematics, Newton's three laws, force and friction, momentum and impulse, collisions, and projectile motion (Content Category 4A, "Translational motion, forces, work, energy"). Numerical practice is the single biggest predictor of your score.

Displacement

Displacement is the shortest straight-line vector from initial to final position. It is a vector quantity with both magnitude and direction. SI unit: meter (m).

Distance, by contrast, is a scalar — the total length of path traveled. A runner completing one lap of a 400 m track has covered 400 m of distance but zero displacement.

Velocity

Velocity is the rate of change of displacement: v = Δs/Δt. It is a vector with SI unit m/s.

Average velocity = total displacement / total time.
Instantaneous velocity = velocity at a single instant, equal to the slope of the displacement-time graph at that point.
Speed is the magnitude of velocity (a scalar).

Acceleration

Acceleration is the rate of change of velocity: a = Δv/Δt. SI unit: m/s². Acceleration is a vector and can be positive (speeding up in the chosen positive direction) or negative (slowing down / deceleration).

Uniform and Variable Acceleration

Uniform acceleration means the velocity changes by equal amounts in equal time intervals. Free-fall under gravity (g = 9.8 m/s², often rounded to 10 m/s² on the MCAT) is the canonical example. The kinematics equations apply only to uniform (constant) acceleration:

v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t

Variable (non-uniform) acceleration occurs when the acceleration itself changes with time. The constant-acceleration kinematics equations no longer apply; you must analyze the motion graphically (slopes and areas) instead.

Displacement-Time Graph

A graph of displacement (y-axis) against time (x-axis) reveals the motion at a glance:

Horizontal line → body at rest.
Straight inclined line → uniform velocity. Slope = velocity.
Curved (concave-up) line → accelerating.
Curved (concave-down) → decelerating.

The slope of the displacement-time graph at a point is the instantaneous velocity at that instant. For a velocity-time graph the slope gives acceleration and area under the graph gives displacement.

Newton's Laws of Motion

Newton's three laws — statement, equation, everyday example
Law	Statement	Equation	Defines / explains	Everyday example
1st (inertia)	A body stays at rest or in uniform motion unless a net external force acts on it.	If F_net = 0 → a = 0	Inertia; concept of inertial frame	Passenger lurches forward when a bus stops suddenly
2nd (acceleration)	Net force equals rate of change of momentum.	F = dp/dt = ma (constant mass)	Quantitative link between force, mass, and acceleration	Pushing an empty cart vs a loaded cart with the same force
3rd (action-reaction)	Every action has an equal and opposite reaction; the two forces act on different bodies.	F_AB = −F_BA	Pair-wise force interaction	Rocket propulsion, walking, recoil of a gun, swimming

Common trap. "Action-reaction pairs cancel each other out" — FALSE. They act on different bodies, so they never cancel. A book resting on a table: the book pushes down on the table (action) and the table pushes up on the book (reaction).

Newton's Second Law and Linear Momentum

Linear momentum: p = mv (vector, SI unit kg·m/s). Newton's second law in its most general form is F = dp/dt. For a constant mass body this reduces to F = ma.

Impulse J = F·Δt = Δp — the change in momentum produced by a force acting for time Δt. SI unit: N·s. The area under a force-time graph equals impulse.

Law of conservation of linear momentum: in the absence of external forces, the total momentum of an isolated system is conserved.

Newton's Third Law

Examples of action-reaction pairs:

Walking: foot pushes ground backward, ground pushes foot forward.
Rocket propulsion: hot gases pushed downward, rocket pushed upward.
Swimming: hands push water back, water pushes swimmer forward.
Recoil of a gun: bullet pushed forward, gun pushed back into shoulder.

Weight, Normal Force and Friction

Mass (kg) is the amount of matter in a body and is the same everywhere. Weight is the gravitational force on that mass: W = mg (a vector in newtons, N), so weight changes with location. The normal force N is the contact force a surface exerts perpendicular to itself; on a horizontal surface with no vertical acceleration, N = mg, but on an incline N = mg·cosθ.

Friction opposes relative motion (or the tendency toward it) between surfaces in contact and always acts parallel to the surface:

Static friction prevents motion up to a maximum: f_s ≤ μ_sN. It is self-adjusting, matching the applied force until the object breaks loose.
Kinetic (sliding) friction acts once motion has started: f_k = μ_kN, with a roughly constant magnitude independent of speed.
For a given pair of surfaces μ_s > μ_k — it takes more force to start sliding than to keep it sliding.
The coefficient of friction μ is dimensionless and does not depend on the contact area.

Inclined plane (angle θ, frictionless first). Resolve weight into components along and perpendicular to the surface: the component down the slope is mg·sinθ and the component into the surface is mg·cosθ. So the acceleration of a block sliding down a frictionless incline is a = g·sinθ (independent of mass). With friction, the net force down the slope is mg·sinθ − μ_kmg·cosθ.

Common trap. The normal force is not always equal to mg. If the surface is inclined, or if an applied force has a vertical component, or if the surface accelerates vertically (e.g. in an elevator), you must recompute N — and since f = μN, the friction changes too.

Momentum and Explosive Forces

An explosion is a process in which an object initially at rest breaks apart. Total momentum before = 0, so total momentum after must also be 0; the fragments fly apart with equal and opposite momenta.

Recoil example: a 4 kg gun fires a 0.02 kg bullet at 400 m/s. By conservation: m_gV = m_bv ⇒ V = (0.02 × 400)/4 = 2 m/s backwards.

Collision (Elastic and Inelastic)

Elastic vs Inelastic vs Perfectly inelastic collision
Property	Elastic	Inelastic	Perfectly inelastic
Momentum conserved?	Yes	Yes	Yes
Kinetic energy conserved?	Yes	No (some converted to heat / sound / deformation)	No (maximum loss of KE)
Bodies after collision	Separate	Separate	Stick together → common velocity
Coefficient of restitution e	e = 1	0 < e < 1	e = 0
Examples	Gas molecules, hard billiard balls (approx)	Most everyday collisions, ball bouncing, car crash with rebound	Lump of clay hitting wall, bullet embedding in block, accident with crumple-zone
Final velocity (1-D)	v₁′, v₂′ from elastic formulas	From momentum + e equation	v = (m₁u₁ + m₂u₂) / (m₁ + m₂)

1-D elastic collision formulas

For two bodies moving along a line:

v₁′ = ((m₁ − m₂)u₁ + 2m₂u₂)/(m₁ + m₂)
v₂′ = ((m₂ − m₁)u₂ + 2m₁u₁)/(m₁ + m₂)

Special cases: If m₁ = m₂ the bodies exchange velocities. If a small body strikes a much heavier stationary body, it rebounds with almost the same speed.

Projectile Motion (Height, Range, Time of Flight, Max Angle)

A projectile is a body thrown into the air with an initial velocity and subsequently moving under gravity alone (air resistance neglected). Horizontal and vertical motions are independent.

For a projectile launched at angle θ with initial speed v:

Time of flight: T = 2v·sinθ/g
Maximum height: H = v²·sin²θ/(2g)
Horizontal range: R = v²·sin(2θ)/g
Maximum range occurs at θ = 45°: R_max = v²/g.
Two angles θ and (90° − θ) give the same range.
Trajectory is a parabola: y = x·tanθ − gx²/(2v²cos²θ).

Memory aid. "Sine for height, sin-2θ for range, 45° is king." Maximum range comes at 45°; same range at complementary angles 30°&60°.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter.

Q1. A body starts from rest and accelerates uniformly at 4 m/s². Its velocity after 5 s is:

9 m/s
10 m/s
15 m/s
20 m/s

Use v = u + at with u = 0: v = 0 + 4 × 5 = 20 m/s.

Q2. A 0.05 kg ball is hit by a bat for 0.01 s, changing its velocity from +20 m/s to −20 m/s. The average force exerted by the bat is:

100 N
150 N
200 N
400 N

Impulse J = Δp = m(v − u) = 0.05 × (−20 − 20) = −2 N·s. Magnitude of force = |J|/Δt = 2/0.01 = 200 N.

Q3. The maximum range of a projectile occurs at an angle of:

30°
45°
60°
90°

R = v²sin(2θ)/g is maximized when sin(2θ) = 1, i.e. 2θ = 90° or θ = 45°.

Q4. In an elastic collision between two equal masses, one moving and the other at rest, after the collision:

Both move with the same velocity
Both come to rest
The moving body stops and the stationary body moves with the original velocity
They stick together

For m₁ = m₂, v₁′ = u₂ = 0 and v₂′ = u₁. Velocities are exchanged — a classic Newton's-cradle outcome.

Q5. A block slides down a frictionless incline that makes a 30° angle with the horizontal. Taking g = 10 m/s², its acceleration down the slope is:

10 m/s²
5 m/s²
8.7 m/s²
2.5 m/s²

On a frictionless incline the net force down the slope is mg·sinθ, so a = g·sinθ = 10 × sin30° = 10 × 0.5 = 5 m/s². The result is independent of the block's mass.

Quick Recap

Kinematics (constant a): v = u + at, s = ut + ½at², v² = u² + 2as.
Weight W = mg; friction f_s ≤ μ_sN, f_k = μ_kN with μ_s > μ_k.
Newton's laws: inertia → F = ma = dp/dt → action-reaction.
Momentum p = mv; impulse J = FΔt = Δp.
Elastic: KE + p conserved; Inelastic: only p conserved.
Projectile: T = 2v·sinθ/g, H = v²sin²θ/(2g), R = v²sin(2θ)/g.
Maximum range at 45°; equal range at θ and 90°−θ.

Test yourself. Take a timed Force and Motion quiz or browse all Physics MCQs to lock these concepts in.