Kinematics: practice problems with step-by-step solutions

Also called: SUVAT equations

Free Kinematics practice problems with diagrams, hints and full worked solutions. New values every time, at three levels: speeding up, braking, vertical throw.

Free practice problemEasy
t (s)v (m/s)9 s6

A train slows down uniformly from 20 m/s to 6 m/s in 9 s. How far does it travel in that time? Find s.

v₀ = 20 m/sv = 6 m/st = 9 s
m
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What you practise

  1. Easyspeeding up
  2. Mediumbraking
  3. Hardvertical throw

How to solve Kinematics problems

  1. List what you know: v₀, v, a, t and s, with units.
  2. Pick the equation that contains your three knowns and the unknown.
  3. Choose one positive direction and give opposing quantities a minus sign.
  4. Solve and check that the answer is physically sensible.

Common mistakes

  • Forgetting to convert km/h to m/s.
  • Leaving out the ½ in s = v₀·t + ½·a·t².
  • Using a positive acceleration for braking.

Key formulas

Velocity
v = v0 + a·t
Displacement
s = v0·t + ½·a·t²
Without time
v² = v0² + 2·a·s
Units
1 m/s = 3.6 km/h

Learn Kinematics from scratch

1What kinematics is

Kinematics describes how things move: how fast, how far and for how long. It does not ask why they move. That is the job of forces and Newton’s laws.

In this topic the acceleration is constant. A car that speeds up steadily, a train that brakes evenly and a stone that falls freely all move like this.

Acceleration is the change in velocity per second. If a car goes from 10 m/s to 13 m/s in one second, its acceleration is 3 m/s² (metres per second, per second).

a = (v − v₀) / t

A negative acceleration means the object is slowing down in the positive direction. Braking is a negative acceleration.

Self-check A car goes from 10 m/s to 16 m/s in 3 s at a steady rate. What is its acceleration?
  1. 18 m/s²
  2. 2 m/s²
  3. 3.3 m/s²
  4. 5.3 m/s²

Answer: 2 m/s². a = (v − v₀)/t = (16 − 10)/3 = 2 m/s². Dividing the final speed alone (16/3 = 5.3) forgets that the car was already moving at 10 m/s.

2The five quantities

Every question in this topic uses some of the same five quantities. Learn their symbols and units first.

  • v₀: the starting velocity, in m/s. “From rest” means v₀ = 0.
  • v: the final velocity, in m/s. “Until it stops” means v = 0.
  • a: the acceleration, in m/s². Constant for the whole motion.
  • t: the time, in s.
  • s: the displacement (distance moved in the positive direction), in m.

Speeds are often given in km/h. Convert them to m/s before you calculate:

1 m/s = 3.6 km/h v (m/s) = v (km/h) / 3.6

Choose one positive direction, usually the direction of motion. A quantity that points the other way gets a minus sign. For a ball thrown upwards with up as positive, gravity gives a = −9.81 m/s².

Self-check A car drives at 54 km/h. What is its speed in m/s?
  1. 15 m/s
  2. 194.4 m/s
  3. 5.4 m/s
  4. 54 m/s

Answer: 15 m/s. Divide by 3.6: 54/3.6 = 15 m/s. Multiplying by 3.6 (194.4) goes the wrong way: a speed in m/s is always a smaller number than the same speed in km/h.

3The three equations

With constant acceleration, three equations connect the five quantities:

v = v₀ + a·t
s = v₀·t + ½·a·t²
v² = v₀² + 2·a·s

Each equation leaves out one quantity. The first has no s, the second has no v and the third has no t. Pick the equation that contains the three things you know and the one you want.

A velocity–time graph shows why the equations work. With constant acceleration the graph is a straight line, and the area under the line is the distance.

t (s)v (m/s)sav0​vt

The area of this shape is the average speed times the time: s = ½·(v₀ + v)·t. This fourth form is handy when a is not given.

Self-check You know v₀, a and s, and you want v. The time is not given. Which equation do you use?
  1. a = (v − v₀)/t
  2. s = v₀·t + ½·a·t²
  3. v = v₀ + a·t
  4. v² = v₀² + 2·a·s

Answer: v² = v₀² + 2·a·s. Only v² = v₀² + 2·a·s contains v₀, a, s and v without t. The other equations all need the time, which you do not have.

4The method step by step

  1. Draw a quick sketch and choose the positive direction.
  2. List v₀, v, a, t and s. Write the value of each known one, with its sign, and mark the unknown.
  3. Convert km/h to m/s and minutes to seconds.
  4. Pick the equation that contains your three knowns and the unknown.
  5. Rearrange, put the numbers in and solve.
  6. Check that the answer makes sense: a braking car does not travel backwards, and a time is never negative.

When the motion has several phases, solve them one after the other. The final velocity of one phase is the starting velocity of the next.

5A worked example

Follow every step. Try to predict the next one before you open it.

Worked example

A car driving at 72 km/h brakes with a constant deceleration of 5 m/s² until it stops. How long does braking take, and what is the braking distance?

1

Convert the speed

v₀ = 72 / 3.6 = 20 m/s
2

List the quantities

Take the direction of motion as positive. The car slows down, so the acceleration is negative.

v₀ = 20 m/s v = 0 a = −5 m/s²

The unknowns are t and s.

3

Find the braking time

Use v = v₀ + a·t, which has no s.

0 = 20 + (−5)·t ⇒ t = 20 / 5 = 4 s
4

Find the braking distance

Use v² = v₀² + 2·a·s, which has no t, so an error in the time cannot carry over.

0 = 20² + 2·(−5)·s ⇒ s = 400 / 10 = 40 m
5

Check the answer

t (s)v (m/s)1234102040 m

The area of the triangle is ½ · 4 s · 20 m/s = 40 m. The average speed is 10 m/s for 4 s, which also gives 40 m.

Answer

Braking takes t = 4 s and the braking distance is s = 40 m.

6Common mistakes

  • Calculating with km/h instead of m/s. Divide by 3.6 first.
  • Leaving out the ½ in s = v₀·t + ½·a·t².
  • Using a positive acceleration for braking. If the motion is positive, braking is negative.
  • Forgetting the minus sign of g for an object thrown upwards when up is positive.
  • Mixing up v₀ and v: “from rest” sets v₀ = 0, “comes to a stop” sets v = 0.
Self-check A cart starts from rest and accelerates at 2 m/s² for 3 s. How far does it travel?
  1. 18 m
  2. 3 m
  3. 6 m
  4. 9 m

Answer: 9 m. s = v₀·t + ½·a·t² = 0 + ½ · 2 · 3² = 9 m. Leaving out the ½ gives 18 m. 6 m is the final speed a·t in m/s, not a distance.

7How exam questions look

  • Easy: speeding up or slowing down. Find a, v or s with one equation, for example a car that speeds up for a given time or a stone dropped from a bridge.
  • Medium: braking and take-off. Speeds in km/h, braking time and distance, a reaction time before braking, or a police car catching up.
  • Hard: vertical throws and motion in phases. A ball thrown up from a roof, two balls that meet, a rocket, or a trip that speeds up, cruises and brakes.

For vertical motion, take up as positive and use a = −9.81 m/s² for the whole flight. At the highest point the velocity is 0 for an instant.

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Worked examples

Example 1Easy
t (s)v (m/s)20 s10

A train slows down uniformly from 15 m/s to 10 m/s in 20 s. How far does it travel in that time? Find s.

v₀ = 15 m/sv = 10 m/st = 20 s

Find s (m)

Show hints
  1. With constant acceleration the average speed is the mean of the initial and final speeds.
  2. s = (v₀ + v)/2 · t. It is the area under the v–t graph.
  3. average speed = 12.5 m/s.
Show the answer and the step-by-step solution

Motion with constant acceleration: v = v₀ + a·t, s = v₀·t + ½·a·t² and v² = v₀² + 2·a·s. Pick the equation that contains the three things you know and the one you want, and keep the signs consistent with one positive direction.

Average speed

(v₀ + v)/2 = (15 + 10)/2 = 12.5 m/s

Distance

s = 12.5·20 = 250 m

Answer

s = 250 m
Example 2Medium
a = ?s = 1000 m

A plane needs a speed of 250 km/h to take off and the usable runway is 1000 m long. Starting from rest, what constant acceleration a does it need?

v₀ = 0v = 250 km/hs = 1000 m

Find a (m/s²)

Show hints
  1. Convert the speed to m/s: divide km/h by 3.6.
  2. v² = v₀² + 2·a·s with v₀ = 0, so a = v²/(2s).
  3. v = 69.44 m/s.
Show the answer and the step-by-step solution

Motion with constant acceleration: v = v₀ + a·t, s = v₀·t + ½·a·t² and v² = v₀² + 2·a·s. Pick the equation that contains the three things you know and the one you want, and keep the signs consistent with one positive direction.

Units

v = 250/3.6 = 69.44 m/s

Acceleration

a = v²/(2s) = 69.44²/(2·1000) = 2.411 m/s²

Answer

a = 2.411 m/s²
Example 3Hard
a = 22 m/s²

A model rocket is launched straight up from rest. Its engine gives it a constant upward acceleration of 22 m/s² for 2 s, after which the engine stops and the rocket moves freely. Ignore air resistance and use g = 9.81 m/s². Find the greatest height hmax the rocket reaches.

a = 22 m/s²t1 = 2 sg = 9.81 m/s²

Find hmax (m)

Show hints
  1. Split the flight: powered phase with acceleration a, then free flight with acceleration −g.
  2. At burnout: v₁ = a·t₁ and h₁ = ½·a·t₁².
  3. v₁ = 44 m/s, h₁ = 44 m.
Show the answer and the step-by-step solution

Motion with constant acceleration: v = v₀ + a·t, s = v₀·t + ½·a·t² and v² = v₀² + 2·a·s. Pick the equation that contains the three things you know and the one you want, and keep the signs consistent with one positive direction.

Powered phase

v₁ = 22·2 = 44 m/s
h₁ = ½·22·2² = 44 m

Free flight up

At the top the velocity is zero.

Δh = v₁²/(2g) = 44²/(2·9.81) = 98.67 m
hmax = 44 + 98.67 = 142.7 m

Answer

hmax = 142.7 m

Questions and answers

What is Kinematics?

Kinematics describes how things move: how fast, how far and for how long. It does not ask why they move. That is the job of forces and Newton’s laws.

In this topic the acceleration is constant. A car that speeds up steadily, a train that brakes evenly and a stone that falls freely all move like this.

How do I solve Kinematics problems step by step?
  1. List what you know: v₀, v, a, t and s, with units.
  2. Pick the equation that contains your three knowns and the unknown.
  3. Choose one positive direction and give opposing quantities a minus sign.
  4. Solve and check that the answer is physically sensible.
What are common mistakes in Kinematics?
  • Forgetting to convert km/h to m/s.
  • Leaving out the ½ in s = v₀·t + ½·a·t².
  • Using a positive acceleration for braking.
What do the Kinematics exercises cover?
  • Easy: speeding up
  • Medium: braking
  • Hard: vertical throw

Every problem has new values, so you learn the method instead of an answer. Stuck? Open a small hint first, and the full explanation only when you need it.

How practising works

  • Every problem has new values, so you learn the method instead of an answer.
  • Stuck? Open a small hint first, and the full explanation only when you need it.
  • Your progress, mistakes and saved problems are kept, so you can review before the exam.

Topics before and after this one

Continue with

Newton’s lawsProjectile motion

Calculators

Kinematics calculator (SUVAT)Free fall calculator (time, height and impact speed)Acceleration calculator (and average velocity)

Courses with this topic

DynamicsPhysics for EngineersPhysics 1: MechanicsAP Physics 1 practice problems

More topics in Physics

Newton’s lawsCircular motionMomentum & collisionsSprings & pendulumsCoulomb’s law & electric fieldsMagnetic fields & inductionWaves, sound & optics

Printable worksheet

Kinematics worksheet with answers
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