Unit 02 - Grade 11-12 Physics

Kinematics

Learn how to describe motion using position, displacement, velocity, acceleration, motion graphs, constant-acceleration equations, and free-fall reasoning.

Lesson roadmap

What Students Should Master in This Unit

Kinematics is the language of motion. It explains where an object is, how far it moved, how fast it is moving, whether it is speeding up or slowing down, and how to read motion from equations and graphs.

Describe motion

Use position, displacement, distance, speed, velocity, and acceleration correctly.

Analyze graphs

Interpret position-time, velocity-time, and acceleration-time graphs using slope and area.

Solve motion problems

Apply constant-velocity, constant-acceleration, and free-fall formulas with clear units.

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Jump to a Topic

Motion Vocabulary Reference Frames Distance and Displacement Speed and Velocity Acceleration Motion Graphs Constant Velocity Kinematic Equations Free Fall Lab Skills Worked Examples Practice Problems
Core language

1. Motion Vocabulary

Before using formulas, students must know what each quantity means. Many wrong answers happen because students confuse distance with displacement or speed with velocity.

Kinematics vocabulary diagram Kinematics describes motion along a chosen axis x_i x_f displacement v a
Position tells where an object is. Displacement tells how position changes, while velocity and acceleration include direction.
Quantity Symbol Meaning Scalar or Vector Common Unit
PositionxLocation relative to an origin.Vector-like with sign in one dimensionm
DistancedTotal path length traveled.Scalarm
DisplacementΔxChange in position: final position minus initial position.Vectorm
Time intervalΔtElapsed time.Scalars
SpeedvHow fast distance is covered.Scalarm/s
VelocityvRate of change of position with direction.Vectorm/s
AccelerationaRate of change of velocity.Vectorm/s2
Student habit: If a question asks for a vector, include direction or a sign. For example, -6.0 m/s can mean 6.0 m/s in the negative direction.
Direction setup

2. Reference Frames and Sign Conventions

A reference frame is the viewpoint or coordinate system used to describe motion. In one-dimensional motion, students usually choose one direction as positive and the opposite direction as negative.

Reference frame and sign convention diagram Choose positive direction before solving positive direction negative direction origin
The sign of velocity and acceleration depends on the positive direction you choose at the start.

How to Choose Signs

  • Choose an origin, such as a starting line, table edge, or ground level.
  • Choose a positive direction, such as east, right, upward, or forward.
  • Keep the sign convention consistent for the whole problem.
  • Use the sign of velocity to show direction of motion.
  • Use the sign of acceleration to show direction of acceleration, not necessarily whether the object is speeding up.
Important idea: Negative acceleration does not always mean slowing down. An object slows down when velocity and acceleration point in opposite directions.
Position change

3. Distance and Displacement

Distance and displacement can be very different. Distance is the total path length. Displacement is the straight-line change from start to finish, including direction.

Distance versus displacement diagram Distance follows the path; displacement connects start to finish distance = 8 m + 3 m displacement = 5 m east start finish
A path can be long even when the final position is close to the starting point.
Displacement Δx = xf - xi Final position minus initial position.
Total distance d = sum of all path lengths Distance is never negative.
Position form xf = xi + Δx Useful when finding final location.

Example Idea

If a student walks 8 m east and then 3 m west, the distance is 11 m, but the displacement is 5 m east.

Rate of motion

4. Speed and Velocity

Speed tells how fast something moves. Velocity tells how fast position changes in a particular direction. Average velocity depends only on displacement and elapsed time, not on the path taken.

Speed and velocity diagram Speed uses distance; velocity uses displacement total path length straight-line displacement start finish
Average speed can be larger than average velocity because it uses total distance instead of displacement.
Average speed average speed = total distance / total time Scalar, always nonnegative.
Average velocity vavg = Δx / Δt Vector, uses displacement.
Constant velocity position x = x0 + vt Works only when velocity is constant.

Instantaneous Velocity

Instantaneous velocity is velocity at one moment. On a position-time graph, it is the slope of the tangent line at that moment. In many Grade 11-12 problems, velocity is treated as constant over a small time interval.

Changing velocity

5. Acceleration

Acceleration measures how quickly velocity changes. Velocity can change because speed changes, direction changes, or both. In one-dimensional kinematics, acceleration is positive or negative depending on the chosen coordinate direction.

Acceleration diagram Acceleration describes how velocity changes acceleration vi vf
If velocity and acceleration point in the same direction, the object speeds up. If they point opposite directions, it slows down.
Average acceleration aavg = Δv / Δt Change in velocity divided by time.
Velocity change Δv = vf - vi Final velocity minus initial velocity.
Velocity with constant acceleration vf = vi + at Use when acceleration is constant.

Speeding Up vs. Slowing Down

Velocity Sign Acceleration Sign Motion Description
PositivePositiveMoving in the positive direction and speeding up.
PositiveNegativeMoving in the positive direction and slowing down.
NegativeNegativeMoving in the negative direction and speeding up.
NegativePositiveMoving in the negative direction and slowing down.
Visual motion

6. Motion Graphs

Motion graphs are one of the most important parts of kinematics. Students should be able to move between descriptions, tables, graphs, equations, and calculations.

Motion graphs diagram Graph shape depends on what the axes represent position-time slope = velocity velocity-time area = displacement
On an x-t graph, slope gives velocity. On a v-t graph, slope gives acceleration and area gives displacement.

Graph Meaning Table

Graph Slope Means Area Means Common Shape
Position vs. timeVelocityUsually not usedStraight line for constant velocity, curve for acceleration.
Velocity vs. timeAccelerationDisplacementHorizontal line for constant velocity, sloped line for constant acceleration.
Acceleration vs. timeUsually not used in basic kinematicsChange in velocityHorizontal line for constant acceleration.

Graph Interpretation Rules

  • A horizontal position-time graph means the object is at rest.
  • A straight sloped position-time graph means constant velocity.
  • A curved position-time graph means changing velocity.
  • A horizontal velocity-time graph means constant velocity and zero acceleration.
  • A sloped velocity-time graph means acceleration.
  • Area above the time axis on a velocity-time graph gives positive displacement.
  • Area below the time axis on a velocity-time graph gives negative displacement.
Exam habit: Never say "the graph is going up, so the object is moving up" unless the vertical axis is position. Always check what the y-axis represents.
Uniform motion

7. Constant Velocity Motion

Constant velocity means the object covers equal displacements in equal time intervals and its acceleration is zero. The position-time graph is a straight line.

Constant velocity motion diagram Constant velocity means equal spacing in equal time intervals same distance same distance same distance t = 0 t = 1s t = 2s t = 3s
When the spacing is equal for equal times, velocity is constant and acceleration is zero.
Position equation x = x0 + vt Final position equals initial position plus displacement.
Displacement equation Δx = vt Use when velocity is constant.
Velocity from graph v = slope of x-t graph Slope units are meters per second.
Constant acceleration

8. Kinematic Equations

The kinematic equations work when acceleration is constant. They connect displacement, time, initial velocity, final velocity, and acceleration.

Four kinematic equations diagram Four constant-acceleration equations Use these when acceleration is constant. Choose the equation that avoids the missing variable. 1. vf = vi + at Best when displacement is not needed. 2. Δx = vit + 1/2 at2 Best when final velocity is not needed. 3. vf2 = vi2 + 2aΔx Best when time is not given. 4. Δx = [(vi + vf) / 2]t Best when acceleration is not needed directly.
These four equations connect displacement, time, initial velocity, final velocity, and constant acceleration.

Variables

Symbol Meaning Common Unit
x0Initial positionm
xFinal positionm
ΔxDisplacementm
vi or v0Initial velocitym/s
vf or vFinal velocitym/s
aAccelerationm/s2
tTime intervals

Four Core Kinematic Equations

Velocity-time vf = vi + at Best when displacement is not needed.
Displacement with time Δx = vit + (1/2)at2 Best when final velocity is not needed.
No time equation vf2 = vi2 + 2aΔx Best when time is not given or needed.
Average velocity displacement Δx = [(vi + vf) / 2]t Only for constant acceleration.

Useful Supporting Forms

Position form x = x0 + vit + (1/2)at2 Use when initial position is not zero.
Average velocity vavg = (vi + vf) / 2 Only for constant acceleration.

How to Choose an Equation

  1. List the known quantities with signs and units.
  2. Identify the unknown quantity.
  3. Find which quantity is missing from the problem.
  4. Choose the equation that does not include the missing quantity.
  5. Substitute carefully and solve.
Gravity motion

9. Free Fall and Vertical Motion

Free fall is motion where gravity is the only force affecting the object's motion. In basic kinematics, air resistance is ignored. Near Earth's surface, the magnitude of gravitational acceleration is about 9.8 m/s2.

Free fall vertical motion diagram At the top, velocity is zero but acceleration is not v upward a = -g v = 0 at top v downward
Gravity points downward during the entire motion, even at the highest point.
Gravity magnitude g = 9.8 m/s2 Use 10 m/s2 only if your teacher allows estimation.
If upward is positive a = -9.8 m/s2 Gravity points downward.
If downward is positive a = +9.8 m/s2 Use this only if you choose downward as positive.

Important Free-Fall Facts

  • At the top of a vertical throw, velocity is zero for an instant.
  • Acceleration is still downward at the top. It is not zero.
  • If an object returns to the same height, the upward and downward speeds have equal magnitude.
  • Time going up equals time coming down only when the launch and landing heights are the same.
Common mistake: Students often set acceleration equal to zero at the top of the path. The velocity is zero at that instant, but acceleration is still -9.8 m/s2 if upward is positive.
Investigation skills

10. Kinematics Lab Skills

Kinematics labs usually involve measuring position and time, then using graphs to determine velocity or acceleration. The goal is not just collecting numbers; it is using data to describe motion clearly.

Kinematics lab setup diagram Kinematics labs turn position-time data into motion graphs motion sensor cart or object position data over time
Students should connect measured data to graphs, then interpret slope and area physically.

Common Lab Tools

  • Meterstick or measuring tape for position and distance.
  • Stopwatch or photogate for time intervals.
  • Motion sensor for position-time and velocity-time data.
  • Video analysis software for frame-by-frame motion.
  • Cart, ramp, track, ball, or falling object depending on the investigation.

Lab Questions Students Should Answer

  • What is the independent variable?
  • What is the dependent variable?
  • What quantities must be controlled?
  • What does the slope of the graph represent?
  • What does the area under the graph represent?
  • What are the largest sources of uncertainty?
Worked examples

11. Worked Examples

Example 1: Distance vs. displacement

A runner moves 120 m east, then 50 m west. Find distance and displacement.

Distance = 120 m + 50 m = 170 m.

Displacement = 120 m east - 50 m west = 70 m east.

Example 2: Average velocity

A car moves from x = 15 m to x = 95 m in 8.0 s. Find average velocity.

Δx = 95 m - 15 m = 80 m.

vavg = Δx / Δt = 80 m / 8.0 s = 10 m/s.

Example 3: Acceleration

A bicycle speeds up from 3.0 m/s to 11.0 m/s in 4.0 s. Find acceleration.

a = Δv / Δt = (11.0 - 3.0) m/s / 4.0 s.

a = 2.0 m/s2.

Example 4: Constant acceleration displacement

A cart starts from rest and accelerates at 1.5 m/s2 for 6.0 s. Find displacement.

Δx = vit + (1/2)at2.

Δx = 0 + (1/2)(1.5)(6.0)2 = 27 m.

Example 5: No time equation

A car traveling 12 m/s accelerates at 2.0 m/s2 over 50 m. Find final speed.

vf2 = vi2 + 2aΔx.

vf2 = 122 + 2(2.0)(50) = 344.

vf = 18.5 m/s.

Example 6: Free fall from rest

A ball is dropped from rest and falls for 3.0 s. Ignore air resistance. Find its velocity after 3.0 s and displacement.

Choose downward as positive: a = 9.8 m/s2, vi = 0.

vf = vi + at = 0 + (9.8)(3.0) = 29.4 m/s downward.

Δy = (1/2)at2 = (1/2)(9.8)(3.0)2 = 44.1 m downward.

Independent practice

12. Practice Problems

Try these first, then open the answer check. Write knowns, unknowns, equation, substitution, and final answer with units.

1. A student walks 6 m north and then 4 m south. Find distance and displacement.

Answer

Distance = 10 m. Displacement = 2 m north.

2. A car travels 150 m in 5.0 s at constant velocity. Find velocity.

Answer

v = Δx / t = 150 m / 5.0 s = 30 m/s.

3. A runner's velocity changes from 2.0 m/s to 8.0 m/s in 3.0 s. Find acceleration.

Answer

a = (8.0 - 2.0) / 3.0 = 2.0 m/s2.

4. A train starts from rest and accelerates at 0.80 m/s2 for 20 s. Find final velocity.

Answer

vf = vi + at = 0 + (0.80)(20) = 16 m/s.

5. A cyclist moving at 5.0 m/s accelerates at 1.2 m/s2 for 4.0 s. Find final velocity.

Answer

vf = 5.0 + (1.2)(4.0) = 9.8 m/s.

6. A car starts from rest and accelerates at 3.0 m/s2 for 7.0 s. Find displacement.

Answer

Δx = (1/2)(3.0)(7.0)2 = 73.5 m.

7. A skateboarder moves at 10 m/s and slows to 2.0 m/s in 4.0 s. Find acceleration.

Answer

a = (2.0 - 10) / 4.0 = -2.0 m/s2.

8. A ball is thrown upward with an initial velocity of 19.6 m/s. How long until it reaches the top?

Answer

At the top, vf = 0. Choose upward positive, a = -9.8 m/s2.

0 = 19.6 - 9.8t, so t = 2.0 s.

9. A rock is dropped from rest. How far does it fall in 2.5 s?

Answer

Choose downward positive. Δy = (1/2)(9.8)(2.5)2 = 30.6 m downward.

10. A car accelerates from 15 m/s to 25 m/s over 100 m. Find acceleration.

Answer

vf2 = vi2 + 2aΔx.

252 = 152 + 2a(100), so 625 = 225 + 200a.

a = 2.0 m/s2.

11. On a velocity-time graph, the velocity is 6.0 m/s for 8.0 s. Find displacement.

Answer

Area = rectangle = (6.0)(8.0) = 48 m.

12. On a velocity-time graph, velocity increases from 0 to 12 m/s in 4.0 s. Find acceleration.

Answer

Slope = Δv / Δt = 12 / 4.0 = 3.0 m/s2.

13. A position-time graph has a slope of -4.0 m/s. What does this mean?

Answer

The object moves with a velocity of 4.0 m/s in the negative direction.

14. A ball is thrown upward and returns to the same height after 6.0 s. How long did it take to reach the top?

Answer

For same launch and landing height, time up equals time down. Time to top = 3.0 s.

15. A car moves at 20 m/s and brakes with acceleration -5.0 m/s2. How long until it stops?

Answer

0 = 20 + (-5.0)t, so t = 4.0 s.

16. The same car in problem 15 stops in 4.0 s. How far does it travel while braking?

Answer

Δx = [(vi + vf) / 2]t = [(20 + 0) / 2](4.0) = 40 m.

Final review

13. What to Know Before Moving On

  • Distance is path length; displacement is change in position.
  • Speed is scalar; velocity includes direction.
  • Acceleration describes change in velocity, not just speeding up.
  • Slope of a position-time graph is velocity.
  • Slope of a velocity-time graph is acceleration.
  • Area under a velocity-time graph is displacement.
  • Kinematic equations only work directly when acceleration is constant.
  • In free fall near Earth, acceleration is 9.8 m/s2 downward.
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