Kinematics Problems with Step-by-Step Solutions: Easy to Hard

Overview

In basic kinematics, you describe motion without explaining its cause. That means the core job is usually to connect five quantities: displacement s, initial velocity u, final velocity v, acceleration a, and time t. In many courses these are grouped as the SUVAT variables.

For constant acceleration, the main equations are:

• v = u + at
• s = ut + 1/2 at²
• v² = u² + 2as
• s = ((u + v) / 2)t

These equations are powerful, but they only work directly when acceleration is constant over the interval you are analyzing. That is one of the most important checks in any physics motion problem.

Before starting any calculation, use this short kinematics checklist:

1. Choose a positive direction. Write it down.
2. List the known quantities with units.
3. Decide whether acceleration is constant.
4. Identify what is being asked.
5. Pick the equation that links knowns to the unknown.
6. Substitute carefully, including signs.
7. Check whether the answer is physically sensible.

If you need a broader revision page for formulas, see Physics Exam Formula Checklist: What to Memorize vs What to Understand. If graphs are part of your kinematics unit, pair this guide with Graphing in Physics: How to Read Position-Time, Velocity-Time, and Acceleration-Time Graphs.

Checklist by scenario

This section gives you a practical method for common kinematics questions and then shows worked examples from easy to hard.

Scenario 1: Straight-line motion from rest

Use this checklist when: an object starts from rest and accelerates uniformly.

• Set u = 0 if it starts from rest.
• Underline the target variable.
• Choose the equation with the fewest extra unknowns.

Worked example 1: Easy

A bicycle starts from rest and accelerates at 2.0 m/s² for 5.0 s. Find its final velocity and displacement.

Step 1: Write known values.
u = 0 m/s
a = 2.0 m/s²
t = 5.0 s

Step 2: Find final velocity.
Use v = u + at
v = 0 + (2.0)(5.0) = 10 m/s

Step 3: Find displacement.
Use s = ut + 1/2 at²
s = 0 + 1/2 (2.0)(5.0)²
s = 1.0 × 25 = 25 m

Answer: final velocity = 10 m/s, displacement = 25 m.

Quick check: starting from rest with positive acceleration means speed should increase steadily, so a positive final velocity and positive displacement make sense.

Scenario 2: Braking or slowing down

Use this checklist when: velocity and acceleration point in opposite directions.

• Pick a positive direction first.
• If the object is slowing down in the positive direction, acceleration is negative.
• Stopping usually means v = 0.

Worked example 2: Easy to medium

A car moves at 20 m/s and brakes uniformly at 4.0 m/s². How long does it take to stop, and what stopping distance does it cover?

Step 1: Choose signs.
Take forward as positive.
u = 20 m/s
v = 0 m/s
a = -4.0 m/s²

Step 2: Find stopping time.
Use v = u + at
0 = 20 + (-4.0)t
0 = 20 - 4.0t
t = 5.0 s

Step 3: Find stopping distance.
Use v² = u² + 2as
0 = 20² + 2(-4.0)s
0 = 400 - 8s
8s = 400
s = 50 m

Answer: time to stop = 5.0 s, stopping distance = 50 m.

Quick check: distance must be positive. A negative value here would usually mean a sign mistake, not that the car moved backward.

Scenario 3: Free fall

Use this checklist when: gravity is the only significant acceleration.

• Choose upward or downward as positive and stay consistent.
• Near Earth, use g ≈ 9.8 m/s² or your course value.
• If the object is dropped, u = 0.

Worked example 3: Medium

A ball is dropped from a height of 45 m. Ignore air resistance. How long does it take to hit the ground, and how fast is it moving just before impact?

Step 1: Choose signs.
Take downward as positive.
u = 0
s = 45 m
a = 9.8 m/s²

Step 2: Find time.
Use s = ut + 1/2 at²
45 = 0 + 1/2 (9.8)t²
45 = 4.9t²
t² = 45 / 4.9 ≈ 9.18
t ≈ 3.03 s

Step 3: Find impact speed.
Use v = u + at
v = 0 + (9.8)(3.03) ≈ 29.7 m/s

Answer: time ≈ 3.0 s, impact speed ≈ 29.7 m/s downward.

Alternative route: use v² = u² + 2as to get speed directly from height.

Scenario 4: Upward throw and return

Use this checklist when: an object goes up, slows, stops momentarily, then comes down.

• At the highest point, vertical velocity is zero.
• Split the motion if the question asks for several stages.
• Total displacement can be zero even when total distance is not.

Worked example 4: Medium

A ball is thrown vertically upward with an initial speed of 18 m/s. Find the maximum height and the total time to return to the launch point. Ignore air resistance.

Step 1: Choose signs.
Take upward as positive.
u = 18 m/s
a = -9.8 m/s²
At maximum height, v = 0

Step 2: Find maximum height.
Use v² = u² + 2as
0 = 18² + 2(-9.8)s
0 = 324 - 19.6s
19.6s = 324
s ≈ 16.5 m

Step 3: Find time to highest point.
Use v = u + at
0 = 18 - 9.8t
t ≈ 1.84 s

Step 4: Use symmetry for total time.
Time up = time down, so total time ≈ 2(1.84) = 3.68 s

Answer: maximum height ≈ 16.5 m, total time ≈ 3.7 s.

Exam note: symmetry works here because the object returns to the same height and air resistance is ignored.

Scenario 5: Two-stage motion

Use this checklist when: acceleration changes between intervals.

• Do not use one SUVAT equation across the whole motion if acceleration is not constant throughout.
• Solve each stage separately.
• The final velocity of one stage becomes the initial velocity of the next.

Worked example 5: Harder

A train starts from rest, accelerates uniformly at 1.5 m/s² for 8.0 s, then continues at constant velocity for 20 s. Find the total displacement.

Stage 1: Accelerating
u = 0
a = 1.5 m/s²
t = 8.0 s

Final velocity after stage 1:
v = u + at = 0 + (1.5)(8.0) = 12 m/s

Displacement in stage 1:
s₁ = ut + 1/2 at²
s₁ = 0 + 1/2 (1.5)(8.0)²
s₁ = 0.75 × 64 = 48 m

Stage 2: Constant velocity
v = 12 m/s
t = 20 s

Displacement in stage 2:
s₂ = vt = (12)(20) = 240 m

Total displacement
s = s₁ + s₂ = 48 + 240 = 288 m

Answer: total displacement = 288 m.

Scenario 6: Relative motion in one dimension

Use this checklist when: two objects move along the same straight line and you are asked when or where they meet.

• Give both objects positions using the same origin.
• Write a motion equation for each object.
• Set positions equal when they meet.

Worked example 6: Hard

Runner A starts at x = 0 and runs at a constant velocity of 4.0 m/s. Runner B starts 20 m behind A but runs at 6.0 m/s. When does B catch A, and where?

Step 1: Define positions.
Let A start at x = 0, so B starts at x = -20 m.

Step 2: Write position equations.
x_A = 4.0t
x_B = -20 + 6.0t

Step 3: Set positions equal.
4.0t = -20 + 6.0t
20 = 2.0t
t = 10 s

Step 4: Find meeting position.
x = 4.0(10) = 40 m

Answer: B catches A after 10 s at x = 40 m.

If you want more structured problem-solving habits before asking for help, see Physics Homework Help Checklist: What to Try Before You Ask for Help. Students preparing for unit tests may also find AP Physics 1 Practice Problems by Unit and Difficulty useful for wider practice.

What to double-check

Before you box your answer, run through this short review list. It catches a large share of avoidable mistakes in kinematics questions and answers.

• Direction and sign: Did you define positive clearly? Is gravitational acceleration negative or positive in your system?
• Units: Are you mixing km/h with m/s, or minutes with seconds?
• Displacement vs distance: If the object turns around, these are not the same.
• Velocity vs speed: Velocity includes direction. Speed does not.
• Rest conditions: “Starts from rest” means u = 0. “Comes to rest” means v = 0.
• Constant acceleration: Did you use SUVAT only where acceleration stayed constant?
• Reasonable magnitude: Does your answer fit the story? A falling ball should not take 30 seconds to fall a few meters.
• Rounding: Keep extra digits during the calculation, then round at the end.

For students who rely heavily on memorized shortcuts, Physics Formula Triangle Guide: When It Helps and When It Misleads is worth reading. It can help you avoid picking formulas mechanically without checking assumptions.

Common mistakes

Most kinematics errors are not advanced. They come from setup problems. Here are the ones that show up repeatedly in physics homework help and exam prep.

1. Using the wrong sign for acceleration

If upward is positive, then gravity is negative. If downward is positive, then gravity is positive. Students often switch this halfway through a problem.

2. Treating every motion question as a one-step problem

Many physics motion problems contain stages: accelerate, cruise, brake; rise, stop, fall; or one object starts later than another. Break the story into separate intervals.

3. Confusing zero velocity with zero acceleration

At the top of a vertical throw, velocity is zero for an instant, but acceleration is still -9.8 m/s² if upward is positive.

4. Using distance when the formula needs displacement

SUVAT formulas use displacement along the chosen axis. If an object moves out and back, net displacement may be small or zero even though distance traveled is large.

5. Plugging into equations before defining variables

Skipping the “knowns” list makes it easier to confuse u and v, or to forget that one variable belongs to a different stage of motion.

6. Forgetting unit conversions

Convert before solving, not after. A common trap is using acceleration in m/s² with time in minutes or speed in km/h.

7. Ignoring graph meaning

In graph-based kinematics, slope and area matter. The slope of a position-time graph gives velocity. The slope of a velocity-time graph gives acceleration. The area under a velocity-time graph gives displacement. For more practice, see Graphing in Physics: How to Read Position-Time, Velocity-Time, and Acceleration-Time Graphs.

8. Giving an answer with no interpretation

If you calculate v = -12 m/s, the negative sign tells you direction. Do not report just “12 m/s” unless the question asks for speed only.

When to revisit

This page is most useful as a repeat reference, not a one-time read. Come back to it whenever your kinematics work changes in difficulty or format.

• Before a quiz or exam: redo one problem from each scenario without looking at the solution.
• When starting a new mechanics unit: review the checklist so your setup habits stay consistent.
• When you begin graph questions: connect algebraic solutions with motion graphs.
• When your class introduces projectiles: revisit sign conventions and component thinking, since vertical motion still follows the same kinematics ideas.
• When solving labs: return to the double-check section for units, rounding, and reasonableness. For reporting data well, see Lab Report Basics for Physics Students: Structure, Graphs, and Error Analysis and Measurement Uncertainty and Significant Figures in Physics Labs.

To make this practical, try this 10-minute revision routine:

1. Pick one easy, one medium, and one harder kinematics problem.
2. Before calculating, write your positive direction and your known variables.
3. Choose the equation verbally: “I know these three values, so I need the equation containing only one unknown.”
4. Solve neatly and check units.
5. Explain in one sentence what your final answer means physically.

If you are teaching or tutoring, this article also works as a diagnostic checklist. A student who gets the algebra right but the sign wrong has a different problem from a student who cannot identify stages of motion. For misconception patterns across mechanics topics, see Teacher's Guide to Common Physics Misconceptions by Topic.

The simplest way to improve at kinematics is not to memorize more equations. It is to become reliable at setup: define direction, list knowns, choose the right model, and check whether the answer fits the motion. Do that consistently, and even harder SUVAT problems become much more manageable.