Kinematics is the part of physics where many students first learn how to turn a word problem into a clear chain of equations, units, and conclusions. This guide gives you a reusable bank of kinematics problems with step-by-step solutions, arranged from beginner to advanced, so you can practice the core motion ideas that appear across GCSE, A-Level, AP Physics, IB, and introductory college courses. Just as importantly, it gives you a checklist for deciding which equation to use, what to sketch, what to calculate first, and what to verify before you move on.
Overview
If you want to solve motion problems consistently, the main goal is not memorizing as many equations as possible. It is learning a repeatable method. Most kinematics questions become manageable when you identify the motion type, list the known quantities, choose a sign convention, and select equations that match the information given.
At this level, the most common quantities are:
- Displacement s or x: change in position, measured in meters
- Velocity v: rate of change of displacement, measured in m/s
- Acceleration a: rate of change of velocity, measured in m/s²
- Time t: measured in seconds
For motion with constant acceleration, the standard equations are:
- v = u + at
- s = ut + 1/2 at²
- v² = u² + 2as
- s = (u + v)t / 2
Here, u is initial velocity and v is final velocity.
Before any calculation, use this short checklist:
- Decide whether the motion is constant velocity, constant acceleration, or piecewise motion.
- Write down the known values with units.
- Choose positive and negative directions.
- Identify what the question asks for.
- Select the equation that contains the unknown and the known variables only.
- Check whether the answer is physically sensible.
If you need a broader formula reference, see Physics Equations Sheet by Topic: GCSE, A-Level, AP Physics, and Intro College. For exam-specific lists, these are also useful: GCSE Physics Equations List and Rearrangement Guide, A-Level Physics Equations List with Definitions and Unit Checks, and AP Physics Formula Sheet Guide: What Every Equation Means.
Checklist by scenario
This section gives worked physics problems in a practical order. Treat each one as a model solution pattern you can return to when similar questions appear.
Scenario 1: Constant speed in one dimension
Problem: A student walks 120 m along a corridor in 80 s at constant speed. Find the speed.
Checklist:
- Motion type: constant velocity
- Known: s = 120 m, t = 80 s
- Unknown: v
- Use: v = s / t
Solution:
v = 120 / 80 = 1.5 m/s
Answer: The speed is 1.5 m/s.
What to learn from this: If acceleration is zero, you usually need only the definition of velocity, not a full constant-acceleration equation.
Scenario 2: Acceleration from rest
Problem: A bicycle starts from rest and reaches 6.0 m/s in 3.0 s. Find its acceleration.
Checklist:
- Motion type: constant acceleration
- Known: u = 0, v = 6.0 m/s, t = 3.0 s
- Unknown: a
- Use: v = u + at
Solution:
6.0 = 0 + a(3.0)
a = 6.0 / 3.0 = 2.0 m/s²
Answer: The acceleration is 2.0 m/s².
What to learn from this: “Starts from rest” means initial velocity is zero. That phrase appears often and is easy to miss when reading quickly.
Scenario 3: Displacement under constant acceleration
Problem: A car travels with initial velocity 4.0 m/s and accelerates at 1.5 m/s² for 6.0 s. Find the displacement.
Checklist:
- Known: u = 4.0 m/s, a = 1.5 m/s², t = 6.0 s
- Unknown: s
- Use: s = ut + 1/2 at²
Solution:
s = (4.0)(6.0) + 1/2(1.5)(6.0)²
s = 24 + 0.75(36)
s = 24 + 27 = 51 m
Answer: The displacement is 51 m.
What to learn from this: Keep the acceleration term separate until the final substitution. It reduces sign and calculator errors.
Scenario 4: Deceleration to a stop
Problem: A train moving at 20 m/s slows uniformly to rest in 50 s. Find its acceleration.
Checklist:
- Known: u = 20 m/s, v = 0, t = 50 s
- Unknown: a
- Use: v = u + at
Solution:
0 = 20 + a(50)
a = -20 / 50 = -0.40 m/s²
Answer: The acceleration is -0.40 m/s².
What to learn from this: Negative acceleration does not automatically mean “slowing down” in every case. It means acceleration points in the negative direction you chose. In this example, that negative value corresponds to deceleration because the train was moving in the positive direction.
Scenario 5: Using v² = u² + 2as when time is missing
Problem: A ball rolls at 3.0 m/s and accelerates uniformly at 2.0 m/s² over a displacement of 8.0 m. Find the final speed.
Checklist:
- Known: u = 3.0 m/s, a = 2.0 m/s², s = 8.0 m
- Unknown: v
- Time not given
- Use: v² = u² + 2as
Solution:
v² = 3.0² + 2(2.0)(8.0)
v² = 9 + 32 = 41
v = √41 ≈ 6.4 m/s
Answer: The final speed is 6.4 m/s.
What to learn from this: This equation is especially useful when time is not given and not needed.
Scenario 6: Free fall from rest
Problem: A stone is dropped from a height of 45 m. Ignore air resistance. Find the time taken to reach the ground.
Checklist:
- Choose downward as positive, or upward as positive, but stay consistent
- Known: u = 0, s = 45 m, a = g ≈ 9.8 m/s² downward
- Unknown: t
- Use: s = ut + 1/2 at²
Solution:
45 = 0 + 1/2(9.8)t²
45 = 4.9t²
t² = 45 / 4.9 ≈ 9.18
t ≈ 3.03 s
Answer: The stone reaches the ground in about 3.0 s.
What to learn from this: In vertical motion near Earth’s surface, acceleration is usually treated as constant. The main source of error is sign convention, not algebra.
Scenario 7: Object thrown upward
Problem: A ball is thrown vertically upward with speed 18 m/s. Find its maximum height above the release point. Ignore air resistance.
Checklist:
- At maximum height, final velocity v = 0
- Take upward as positive
- Known: u = 18 m/s, v = 0, a = -9.8 m/s²
- Unknown: s
- Use: v² = u² + 2as
Solution:
0 = 18² + 2(-9.8)s
0 = 324 - 19.6s
19.6s = 324
s = 324 / 19.6 ≈ 16.5 m
Answer: The maximum height is 16.5 m.
What to learn from this: “Maximum height” is a clue that instantaneous velocity is zero at the top.
Scenario 8: Average velocity from a velocity-time graph idea
Problem: A runner accelerates uniformly from 2 m/s to 8 m/s in 6 s. Find the displacement during that time.
Checklist:
- Known: u = 2 m/s, v = 8 m/s, t = 6 s
- Unknown: s
- Use: s = (u + v)t / 2
Solution:
s = (2 + 8)(6) / 2 = 10 × 3 = 30 m
Answer: The displacement is 30 m.
What to learn from this: This equation comes from average velocity for constant acceleration. It is often the fastest method.
Scenario 9: Two-stage motion
Problem: A car accelerates from rest at 2.0 m/s² for 5.0 s, then continues at constant velocity for 10.0 s. Find the total displacement.
Checklist:
- Break the problem into intervals
- Stage 1: accelerated motion
- Stage 2: constant velocity using final velocity from stage 1
Solution:
Stage 1:
u = 0, a = 2.0 m/s², t = 5.0 s
v = u + at = 0 + 2.0(5.0) = 10 m/s
s₁ = ut + 1/2 at² = 0 + 1/2(2.0)(5.0)² = 25 m
Stage 2:
Constant velocity v = 10 m/s for 10.0 s
s₂ = vt = 10(10.0) = 100 m
Total displacement:
s = s₁ + s₂ = 25 + 100 = 125 m
Answer: The total displacement is 125 m.
What to learn from this: Many advanced-looking kinematics problems are just simpler segments joined together.
Scenario 10: Relative motion in one dimension
Problem: Two cyclists start 180 m apart and ride toward each other. One moves at 5.0 m/s and the other at 4.0 m/s. How long until they meet?
Checklist:
- Use relative speed when objects move toward each other
- Relative speed = 5.0 + 4.0 = 9.0 m/s
- Use time = distance / relative speed
Solution:
t = 180 / 9.0 = 20 s
Answer: They meet after 20 s.
What to learn from this: Not every motion question needs SUVAT notation. Sometimes a simple distance-rate-time setup is cleaner.
What to double-check
Before you finalize any answer, pause for a quick review. This is where many marks are recovered.
- Units: Convert km/h to m/s if needed. Convert minutes to seconds. A correct formula with mixed units gives a wrong answer.
- Direction: Is upward positive or downward positive? Left or right? State it if the problem could be ambiguous.
- Displacement vs distance: Displacement includes direction; distance does not. Exams often test this distinction directly.
- Velocity vs speed: Speed is scalar. Velocity has direction. If your answer is negative, ask whether the question requested velocity or speed.
- Rest conditions: “Starts from rest” means u = 0. “Comes to rest” means v = 0.
- Constant acceleration assumption: Use SUVAT equations only when acceleration is constant over the interval you are analyzing.
- Reasonableness: A dropped object should not take 20 s to fall a few meters. A bicycle should not reach hundreds of m/s in a school-level problem. Rough physical sense matters.
- Significant figures: Match the precision reasonably to the data given. For a refresher, see Significant Figures Rules in Physics: How to Round, Multiply, and Report Results.
If your problem comes from an experiment rather than a textbook, you may also need to consider uncertainty and measurement quality. In that case, see Uncertainty and Error in Physics Labs: Rules, Examples, and Calculation Methods and Physics Lab Report Checklist: Sections, Graphs, Uncertainty, and Common Mistakes.
Common mistakes
Most errors in kinematics are not caused by difficult mathematics. They come from avoidable setup problems.
1. Using the wrong equation first
Students often reach for a familiar formula instead of the formula that fits the known variables. A better habit is to list the knowns and pick the equation containing only one unknown.
2. Forgetting the sign of acceleration due to gravity
If upward is positive, then gravitational acceleration is negative. If downward is positive, then it is positive. The physics is the same; the algebra changes with your convention.
3. Mixing up displacement and total path length
If a ball moves 3 m right and then 3 m left, the distance traveled is 6 m but the displacement is 0. That distinction is simple in theory and easy to mishandle in timed work.
4. Treating all motion as one stage
Many word problems include acceleration, cruising, braking, or waiting intervals. Solve each interval separately and connect them carefully.
5. Ignoring the meaning of negative answers
A negative displacement or velocity is not automatically wrong. It may simply indicate direction. Check the wording before changing the sign.
6. Algebra slips after a correct setup
Even when the physics is right, rearranging equations can introduce errors. Write each substitution clearly, especially with squared terms and fractions.
7. Not sketching the situation
A quick line diagram or rough velocity-time graph often reveals the correct sign convention and helps you see whether the answer is plausible.
If you are revising for a broader mechanics unit, it can help to pair kinematics with force-based topics after you are confident with motion descriptions alone. A later follow-up topic might be Magnetic Force and Fields: Right-Hand Rules, Formulas, and Solved Problems or circuit motion analogies such as Capacitors and RC Circuits Explained with Charging and Discharging Graphs, depending on your course sequence.
When to revisit
This is a good topic to revisit whenever your inputs change: before an exam block, when you move from one qualification level to another, or when your course starts using graphs, practical data, or multistage motion more heavily.
Come back to this checklist when:
- You begin a new mechanics chapter and want a clean refresher on motion basics.
- You notice repeated mistakes with signs, units, or equation choice.
- You start solving projectile or force problems and need stronger one-dimensional motion foundations.
- You are preparing revision notes for GCSE, A-Level, IB, AP Physics, or intro college exams.
- You switch from straightforward textbook questions to lab-style or graph-based questions.
Practical next-step routine:
- Redo two easy problems without looking at the solutions.
- Choose one vertical motion problem and one multistage problem.
- For each, write the checklist first: motion type, knowns, unknown, sign convention, equation choice.
- After solving, spend one minute checking units, signs, and physical reasonableness.
- Keep your own error log. If you repeatedly miss the same step, add it to your personal pre-solution checklist.
Kinematics gets easier when the process becomes familiar. The equations do matter, but the bigger skill is learning to identify structure: what is known, what is changing, and which equation matches that situation. Return to these worked examples as a reference set, then expand them by changing the numbers, reversing directions, or splitting the motion into more stages. That is how a short list of formulas becomes a real problem-solving toolkit.
For structured revision beyond kinematics, you may also find these guides useful: IB Physics Revision Guide: Topic-by-Topic Formula and Concept Checklist, GCSE Physics Equations List and Rearrangement Guide, A-Level Physics Equations List with Definitions and Unit Checks, and AP Physics Formula Sheet Guide: What Every Equation Means.
