Kinematics

What this unit covers

The topics below follow the published Physics 1 course framework for Unit 1. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Position, Displacement & Velocity12 min · 3 objectivesDistinguish position, distance, and displacement using a sign convention · Compute displacement as a change in position, Δx = x_f − x_i · Calculate average velocity and contrast it with average speed
• Acceleration & the Kinematic Equations14 min · 3 objectivesDefine acceleration as the rate of change of velocity · Select the correct kinematic equation from the known and unknown quantities · Solve constant-acceleration problems, including braking to a stop
• Reading Motion Graphs13 min · 3 objectivesInterpret the slope of a position–time graph as velocity · Interpret the slope of a velocity–time graph as acceleration · Find displacement from the area under a velocity–time graph
• Projectile Motion14 min · 3 objectivesTreat horizontal and vertical motion as independent · Analyze a horizontally launched projectile using time as the shared variable · Explain why horizontal velocity stays constant while vertical velocity changes
• Graph Translation & Non-Uniform Acceleration16 min · 3 objectivesConvert between position–time, velocity–time and acceleration–time graphs by slope and area · Read a curved velocity–time graph, where the kinematic equations no longer apply · Extract a physical quantity from the slope of a deliberately linearized plot
• Projectiles Launched at an Angle15 min · 3 objectivesResolve a launch velocity into independent horizontal and vertical components · Use the symmetry of the parabolic path to shortcut time-of-flight and speed questions · Derive the range equation and explain why 45° maximizes range on level ground

Formulas in Unit 1

Every term in Unit 1

All 52 terms we publish for Kinematics, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Displacement vs distance

Displacement is the straight-line change in position and can be zero on a round trip; distance is total path length and never decreases.

Average vs instantaneous velocity

Average velocity is displacement over elapsed time; instantaneous velocity is the slope of the position-time graph at one instant.

Kinematic equations condition

v = v₀ + at, x = x₀ + v₀t + ½at², v² = v₀² + 2aΔx. All require CONSTANT acceleration; they are wrong otherwise.

Position-time graph

Slope is velocity. A curved graph means changing velocity, so acceleration is non-zero.

Velocity-time graph

Slope is acceleration and the area under the curve is displacement. Area below the axis counts as negative displacement.

Projectile motion independence

Horizontal and vertical motions are independent, sharing only the time. Horizontal velocity is constant; vertical acceleration is g.

Time of flight for a projectile

Determined entirely by the vertical motion. A ball thrown horizontally and one dropped from the same height land together.

Relative motion

Velocity depends on the reference frame. v_AC = v_AB + v_BC — velocities add as vectors, not as numbers.

Vector components

A vector at angle θ has components v cos θ along x and v sin θ along y. Adding vectors means adding components separately.

Reading a graph to find another quantity

Slope gives the derivative and area gives the integral: position → velocity → acceleration by slope, and back by area.

Linearizing data

Rewrite the relationship so it has the form y = mx + b, then plot those quantities. Plotting d against t² for free fall gives a straight line of slope ½g.

Slope with units

Always state what the slope of a graph physically represents and its units. A velocity-time slope of 2.4 is 2.4 m/s², not just "2.4".

Best-fit line vs connecting dots

A best-fit line averages out random error; connecting points treats every measurement as exact and hides the trend.

Systematic vs random error

Systematic error shifts every measurement the same way and does not shrink with repetition; random error scatters and does shrink when results are averaged.

Identifying the independent variable

The quantity you deliberately change; it belongs on the horizontal axis. The dependent variable is what you measure in response.

Control variable

A quantity deliberately held constant so it cannot explain the observed change. Naming one is worth a point on nearly every design question.

Reducing uncertainty in timing

Time many cycles and divide, rather than timing one. Human reaction error is a fixed amount, so spreading it over ten cycles cuts its effect tenfold.

Instantaneous velocity from a position graph

Draw a tangent at the instant and take its slope. The secant slope between two nearby points approximates it.

Sign conventions

Choose a positive direction and use it consistently. A negative acceleration means "toward the negative direction", not necessarily "slowing down".

Interpreting a curved velocity-time graph

Curvature means acceleration is changing, so the kinematic equations do not apply and area must be estimated geometrically.

Scalar vs vector

A scalar has magnitude only — distance, speed, time, mass, energy. A vector has magnitude and direction — displacement, velocity, acceleration, force, momentum. Adding them differently is the first place answers diverge.

Distance vs displacement

Distance is path length and never decreases. Displacement is the straight-line change in position and can be zero after a round trip. A lap of a track has large distance and zero displacement.

Average velocity vs average speed

Average velocity is displacement over time; average speed is distance over time. They differ whenever the motion reverses, and the exam picks exactly those cases.

Instantaneous velocity

The slope of the tangent to a position-time graph at one instant, or the limit of average velocity over a vanishing interval.

Acceleration

Rate of change of velocity, a = Δv / Δt. A vector — an object slowing down has acceleration opposite to its velocity, not negative acceleration in some absolute sense.

When speed increases or decreases

Speed increases when velocity and acceleration point the same way, and decreases when they oppose. Both can be negative and the object still speeds up.

Slope of a position-time graph

Velocity. A straight line means constant velocity; a curve means acceleration; a horizontal line means at rest.

Slope of a velocity-time graph

Acceleration. A horizontal line means constant velocity and zero acceleration.

Area under a velocity-time graph

Displacement, with area below the axis counting as negative. This is how a graph question about "how far from the start" is answered.

Area under an acceleration-time graph

Change in velocity, not velocity itself. You need the initial velocity to get the final one.

Kinematic equation without displacement

v = v₀ + at. Choose it when distance is neither given nor wanted.

Kinematic equation without final velocity

Δx = v₀t + ½at². Choose it when the final velocity is neither given nor wanted.

Kinematic equation without time

v² = v₀² + 2aΔx. The one to reach for when the problem never mentions how long.

Choosing a kinematic equation

List what you have and what you want, then pick the equation missing the quantity you neither have nor need. Doing this first saves solving simultaneous equations.

Free fall

Motion under gravity alone, with a = 9.8 m/s² downward regardless of mass. Air resistance is neglected unless the problem says otherwise.

Velocity at the top of a vertical throw

Zero, but the acceleration is still 9.8 m/s² downward. "Zero velocity means zero acceleration" is the single most common wrong answer in this unit.

Symmetry of projectile flight

For equal launch and landing heights, time up equals time down, and the speed at any height going up equals the speed at that height coming down.

Independence of projectile components

Horizontal and vertical motion are independent. Horizontal velocity is constant; vertical velocity changes at g. Time is the only quantity they share.

Time of flight for a horizontal launch

Set by the fall height alone: h = ½gt², so t = √(2h/g). Launch speed does not affect it.

Range of a projectile

Horizontal velocity times time of flight. Since horizontal velocity is constant, range is a multiplication once you have the time.

Projectile launched at an angle

Resolve into v₀cosθ horizontally and v₀sinθ vertically, then treat the two independently. Never use the full speed in a vertical equation.

Reference frame

Motion is described relative to a chosen frame, and velocities add as vectors between frames. A person walking forward on a moving train has one velocity relative to the train and another relative to the ground.

Sign conventions in one dimension

Choose a positive direction at the start and keep it for the whole problem. Most kinematics errors are a sign flipped halfway through, not a wrong formula.

Reading a curved position-time graph

Increasing steepness means speeding up; decreasing steepness means slowing down. Concavity gives the sign of the acceleration.

Why (v0 + v)/2 is only sometimes the average velocity

That shortcut is the mean of the endpoints, which equals the true average only when acceleration is CONSTANT. On a curved velocity-time graph the average is the area divided by the time.

Zero velocity does not mean zero acceleration

At the top of a vertical throw v = 0 while a = -g throughout. Velocity being momentarily zero says nothing about how fast it is changing.

Speed is never negative

Velocity carries a sign for direction; speed is its magnitude. A velocity of -12 m/s is a speed of 12 m/s, and a negative speed is always an arithmetic slip.

Negative area under a velocity-time graph

Area below the axis is displacement in the negative direction. Adding areas with their signs gives displacement; adding their magnitudes gives distance.

A horizontal line on each motion graph means something different

Flat position means at rest; flat velocity means constant velocity; flat acceleration means uniform acceleration. The same shape, three different claims.

The two roots of a projectile time equation

Solving for when a projectile is at a given height usually gives two times, on the way up and on the way down. Discard a negative root; keep both positive ones unless the question rules one out.

Relative velocity in one dimension

v of A relative to B is v_A - v_B. Two cars at 30 and 25 m/s in the same direction approach at 5 m/s; head-on they close at 55 m/s.

Air resistance breaks projectile symmetry

The idealized parabola has equal rise and fall times and equal launch and landing speeds. With drag the descent takes longer and the landing speed is smaller, so the symmetry arguments stop working.

What examiners penalize here

• On the AP exam, always state your positive direction before plugging in numbers. Half of kinematics is bookkeeping — a consistent sign convention earns points that raw algebra cannot.
• When a problem gives you velocities and a distance but never mentions time, reach straight for v² = v₀² + 2aΔx. Recognizing that missing variable saves you from solving a needless quadratic.
• Free-response graph questions love the chain "slope of x–t → v, slope of v–t → a, area under v–t → Δx". Memorize those three links and you can convert between any pair of graphs.
• The classic AP trap: applying gravity to the horizontal axis. Horizontal velocity is constant for every projectile. Gravity changes only the vertical velocity. Keep the two axes in separate columns on your paper.
• If a free-response part says "the students plot the data so that the graph is linear", it is asking you to rearrange the equation into y = mx + b and name the axes. State what goes on each axis *and* what the slope represents — both are separate rubric points.
• Write "up is positive" at the top of any projectile free-response and use it consistently. Rubrics award the sign convention, and a single flipped sign in the vertical equation usually propagates into every later part of the question.

Practice Physics 1

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 8 units of AP Physics 1: Algebra-Based

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations
8. Unit 8 · Fluids