Some quantities need a direction as well as a size. Vectors let you add up journeys and forces correctly.
➡️ Scalars, vectors and displacement
A scalar has size only: distance, speed, mass, time, energy. A vector has size and direction: displacement, velocity, acceleration, force. Distance is the length of the path travelled; displacement is the straight-line change in position, from start to finish.
walk 3 m east, then 3 m west
distance = 3 + 3 = 6 m
displacement = 0 m (back at the start)
run round a 400 m track once
distance = 400 m, displacement = 0 m
🖼️ See it
Perpendicular vectors are added head-to-tail; the resultant is the hypotenuse.
➕ Adding vectors
Vectors in the same direction add; in opposite directions they subtract. At right angles, draw them head-to-tail and use Pythagoras for the size of the resultant: R² = a² + b².
4 N east + 3 N east = 7 N east
4 N east + 3 N west = 1 N east
3 m east then 4 m north
R² = 3² + 4² = 9 + 16 = 25 → R = 5 m
Always give a direction with a vector answer: “5 m” is a distance, “5 m in a direction 37° north of east” is a displacement.
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Scalar or vector
Which of these is a vector?
Does it point somewhere?
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Scalar or vector
Which of these is a scalar?
No direction needed.
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Distance vs displacement
A hiker walks 5 km north then 5 km south. Her displacement is:
Compare start and finish.
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Opposite forces
Forces of 10 N east and 4 N west act on a box. The resultant is:
Subtract the smaller.
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A walker goes 3 m east then 4 m north.
How far is she from the start in a straight line, in m?
Enter an exact answer, without rounding.
m
Use Pythagoras.
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A runner completes 2.5 laps of a 400 m track.
What distance did the runner cover, in m?
Enter an exact answer, without rounding.
m
Laps × length of one lap.
✍️ Type your answer · 7 of 7
Two forces of 6 N and 8 N act at right angles on an object.
What is the size of the resultant force, in N?
Enter an exact answer, without rounding.
N
Right angles mean Pythagoras.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Displacement is:
Question 2 of 5
Which pair are both scalars?
Question 3 of 5
A 7 N force east and a 7 N force west act on a crate. The net force is:
Question 4 of 5
To find the resultant of two perpendicular vectors you use:
Question 5 of 5
After a full lap of a circular track, a runner’s displacement is:
Lesson 2 · Kinematics
🚗 Equations of Motion
When acceleration is constant, a few equations link speed, time and distance, so you can predict a motion before it happens.
📐 The constant-acceleration equations
For constant acceleration a, starting at speed u and reaching speed v after time t, covering distance s:
v = u + at · s = (u + v)/2 × t · s = ut + ½at² · v² = u² + 2as
car from rest, a = 3 m/s², t = 5 s
v = 0 + 3 × 5 = 15 m/s
s = ½ × 3 × 5² = 37.5 m
no time given? use v² = u² + 2as
u = 0, a = 2, s = 100 → v² = 400 → v = 20 m/s
🖼️ See it
The slope is the acceleration (3 m/s²) and the shaded area is the distance travelled, ½ × 5 × 15 = 37.5 m.
🛑 Braking and choosing the right equation
List what you know (u, v, a, s, t), find what you want, and pick the equation that has exactly those four. A braking car has a negative acceleration, and it stops when v = 0.
braking from 20 m/s to 0 over 40 m
0 = 20² + 2 × a × 40
0 = 400 + 80a → a = −5 m/s²
time to stop: t = (v − u)/a = (0 − 20)/(−5) = 4 s
Minus signs matter. Slowing down gives a negative acceleration, not a negative speed.
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Which equation
You know u, v and a and want t. Which equation do you use?
Look for the one with those four letters.
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Stopping
When a car comes to a stop, its final speed v is:
At rest.
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Slowing down
A car that is slowing down has an acceleration that is:
Think about direction.
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Constant acceleration
These equations only work when:
What stays the same?
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A car starts from rest and accelerates at 3 m/s² for 5 s.
What is its final speed, in m/s?
Enter an exact answer, without rounding.
m/s
v = u + at.
✍️ Type your answer · 6 of 7
A cyclist starts from rest and accelerates at 2 m/s² for 6 s.
How far does she travel, in m?
Enter an exact answer, without rounding.
m
s = ut + ½at² with u = 0.
✍️ Type your answer · 7 of 7
A car brakes from 20 m/s to rest in 4 s.
What is the size of its deceleration, in m/s²?
Enter an exact answer, without rounding.
m/s²
Change in speed ÷ time.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
A train accelerates from 10 m/s to 30 m/s in 10 s. Its acceleration is:
Question 2 of 5
A ball starts from rest and has a = 4 m/s² for 3 s. The distance is:
Question 3 of 5
Using v² = u² + 2as with u = 0, a = 5 m/s² and s = 10 m gives v =
Question 4 of 5
Which equation has no time in it?
Question 5 of 5
A car moving at 12 m/s accelerates at 2 m/s² for 4 s. Its final speed is:
Lesson 3 · Gravity
🎯 Free Fall & Projectiles
Anything thrown or dropped accelerates downwards at the same rate. Splitting a throw into horizontal and vertical motion makes projectiles easy.
🍎 Free fall
Ignoring air resistance, every falling object has the same acceleration, g = 10 m/s² downwards, whatever its mass. Use the equations of motion with a = g. For something thrown up, the speed falls by 10 m/s every second until it reaches zero at the top, then it falls back.
dropped from rest for 3 s
v = gt = 10 × 3 = 30 m/s
h = ½gt² = ½ × 10 × 9 = 45 m
thrown up at 20 m/s
time to the top = 20 / 10 = 2 s
max height = 20² / (2 × 10) = 20 m
🖼️ See it
Horizontally the ball moves at a steady 6 m/s; vertically it accelerates at g, so it lands 12 m away after 2 s.
🏹 Projectile motion
A projectile moves in two independent ways at once. Horizontally the speed is constant (no horizontal force). Vertically it accelerates downwards at g. The two motions share only the time of flight.
ball rolls off a 20 m cliff at 6 m/s
fall time: 20 = ½ × 10 × t² → t² = 4 → t = 2 s
horizontal distance = 6 × 2 = 12 m
A dropped ball and a ball fired sideways from the same height land together. Sideways speed does not change how fast you fall.
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Mass and falling
Ignoring air resistance, a heavy stone and a light stone dropped together:
Acceleration does not depend on mass.
✏️ Quick check · 2 of 7
Top of the throw
At the very top of its path, a ball thrown straight up has:
It is about to fall.
✏️ Quick check · 3 of 7
Horizontal motion
The horizontal speed of a projectile (no air resistance) is:
No horizontal force.
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Independence
A ball is dropped and another is fired sideways from the same height. Which lands first?
Compare vertical motion only.
✍️ Type your answer · 5 of 7
A stone is dropped from rest (g = 10 m/s²).
What is its speed after 4 s, in m/s?
Enter an exact answer, without rounding.
m/s
v = gt.
✍️ Type your answer · 6 of 7
A stone is dropped from rest from a bridge and falls for 3 s.
How far does it fall, in m?
Enter an exact answer, without rounding.
m
h = ½gt².
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A ball is thrown straight up at 30 m/s.
How long does it take to reach the top, in s?
Enter an exact answer, without rounding.
s
The speed drops by g each second.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
The acceleration of a freely falling object near the Earth is about:
Question 2 of 5
A ball falls for 2 s from rest. The distance fallen is:
Question 3 of 5
A ball is fired horizontally at 5 m/s and takes 3 s to land. It lands how far away?
Question 4 of 5
Why do a feather and a hammer fall together on the Moon?
Question 5 of 5
For a projectile, which quantity links the horizontal and vertical motion?
Lesson 4 · Forces
🧱 Friction & Springs
Friction opposes sliding, and springs push back in proportion to how far they are stretched. Both change the net force on an object.
🛷 Friction
Friction acts along the surfaces in contact and opposes the motion (or the tendency to move). It depends on how rough the surfaces are and how hard they are pressed together. Static friction stops an object starting to slide and is usually larger than sliding friction. The net force is the driving force minus friction, and F = ma then gives the acceleration.
pushing force 40 N, friction 15 N
net force = 40 − 15 = 25 N
mass 5 kg → a = 25 / 5 = 5 m/s²
constant speed → net force = 0 → friction equals the push
🖼️ See it
A straight line through the origin shows force is proportional to extension; the slope is the spring constant.
🌀 Hooke’s law
A spring’s force is proportional to its extension: F = kx, where k is the spring constant (N/m) and x is the extension from its natural length. This holds only up to the limit of proportionality; beyond it the spring stays permanently stretched. Elastic potential energy stored: E = ½kx².
k = 200 N/m, stretched 0.05 m
F = 200 × 0.05 = 10 N
E = ½ × 200 × 0.05² = ½ × 200 × 0.0025 = 0.25 J
Use the extension, not the total length. A 12 cm spring stretched to 15 cm has x = 3 cm.
✏️ Quick check · 1 of 7
Friction direction
Friction on a sliding box acts:
It is a resistance.
✏️ Quick check · 2 of 7
Constant speed
A crate slides at constant speed when pushed with 30 N. The friction is:
Balanced forces.
✏️ Quick check · 3 of 7
Hooke’s law
Hooke’s law says the force from a spring is proportional to its:
How far it stretches.
✏️ Quick check · 4 of 7
Reducing friction
Which reduces friction between two surfaces?
Think of oiling a bicycle chain.
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A 5 kg box is pushed with 40 N and friction is 15 N.
What is its acceleration, in m/s²?
Enter an exact answer, without rounding.
m/s²
Find the net force first.
✍️ Type your answer · 6 of 7
A spring has k = 200 N/m and is stretched by 0.05 m.
What force does it exert, in N?
Enter an exact answer, without rounding.
N
F = kx.
✍️ Type your answer · 7 of 7
A spring is stretched 4 cm by a force of 8 N.
What is the spring constant, in N/m?
Enter an exact answer, without rounding.
N/m
Convert 4 cm to metres.
📝 Lesson quiz · 5 questions
0 / 5
Re-read the lesson, then give it another go.
Question 1 of 5
Static friction compared with sliding friction is usually:
Question 2 of 5
A spring with k = 50 N/m has F = 10 N. Its extension is:
Question 3 of 5
A 2 kg trolley has a net force of 6 N. Its acceleration is:
Question 4 of 5
Beyond the limit of proportionality, a spring:
Question 5 of 5
Friction usually transfers kinetic energy to:
Lesson 5 · Energy
🎢 Energy Conservation
Energy is never created or destroyed, only moved between stores. Tracking kinetic and gravitational energy predicts speeds and heights.
⚡ Kinetic and potential energy
Kinetic energy Ek = ½mv². Gravitational potential energy Ep = mgh. Work done = force × distance moved in the direction of the force. Power = energy ÷ time (watts).
2 kg at 3 m/s: Ek = ½ × 2 × 9 = 9 J
5 kg raised 4 m: Ep = 5 × 10 × 4 = 200 J
pushing 20 N through 6 m: work = 20 × 6 = 120 J
in 10 s → power = 120 / 10 = 12 W
🖼️ See it
The two energies always add up to 50 J: potential energy becomes kinetic energy as the ball falls.
🔄 Conservation of energy
With no friction, mechanical energy is constant, so a fall converts Ep into Ek: mgh = ½mv². The mass cancels, so v = √(2gh). With friction or air resistance, some energy becomes thermal energy; the total is still conserved. Efficiency = useful output ÷ total input.
ball dropped 5 m (no air resistance)
v = √(2 × 10 × 5) = √100 = 10 m/s
motor: 200 J in, 150 J useful out
efficiency = 150 / 200 = 0.75 = 75%
Energy is not “used up”. It spreads out into the surroundings, usually as thermal energy that is hard to reuse.
✏️ Quick check · 1 of 7
Kinetic energy
If the speed of an object doubles, its kinetic energy:
Speed is squared.
✏️ Quick check · 2 of 7
Conservation
Energy cannot be:
The law of conservation.
✏️ Quick check · 3 of 7
Pendulum
At the lowest point of a swinging pendulum, the energy is mostly:
It moves fastest at the bottom.
✏️ Quick check · 4 of 7
Wasted energy
The “wasted” energy from most machines ends up as:
Machines get warm.
✍️ Type your answer · 5 of 7
A 2 kg ball moves at 3 m/s.
What is its kinetic energy, in J?
Enter an exact answer, without rounding.
J
E_k = ½mv².
✍️ Type your answer · 6 of 7
A 5 kg box is lifted 4 m (g = 10 m/s²).
What gravitational potential energy does it gain, in J?
Enter an exact answer, without rounding.
J
E_p = mgh.
✍️ Type your answer · 7 of 7
A ball is dropped from 5 m with no air resistance (g = 10 m/s²).
What is its speed just before landing, in m/s?
Enter an exact answer, without rounding.
m/s
mgh = ½mv².
📝 Lesson quiz · 5 questions
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Question 1 of 5
A machine takes in 400 J and gives out 300 J of useful energy. Its efficiency is:
Question 2 of 5
Lifting 10 kg through 2 m gives E_p =
Question 3 of 5
Work done by a 50 N force over 4 m is:
Question 4 of 5
A 100 W bulb transfers how much energy in 10 s?
Question 5 of 5
As a skydiver falls, gravitational potential energy mainly becomes:
Lesson 6 · Electricity
🔌 Current, Voltage & Resistance
Current is a flow of charge, voltage is the push behind it, and resistance is the opposition. Ohm’s law links all three.
🔋 Charge, current and voltage
Current I is the rate of flow of charge: I = Q ÷ t, measured in amperes (1 A = 1 coulomb per second). Voltage (potential difference) V is the energy transferred per coulomb, in volts. Resistance R is measured in ohms. Ohm’s law: V = IR for a fixed resistor.
6 C passes in 3 s → I = 6 / 3 = 2 A
V = IR: 2 A through 5 Ω → V = 10 V
12 V across 4 Ω → I = 12 / 4 = 3 A
🖼️ See it
In series the same current flows through each resistor and the resistances add.Each branch has the full voltage; the total resistance is smaller than either branch.
🧶 Series and parallel
Series: one path, same current everywhere, voltages add, resistances add (R = R₁ + R₂). Parallel: branches share the supply, each branch has the full voltage, currents add, and the total resistance is smaller than any branch. Power P = IV.
series: 3 Ω + 6 Ω = 9 Ω; on 18 V → I = 18 / 9 = 2 A
parallel: two equal 10 Ω resistors → total 5 Ω
lamp 6 V, 0.5 A → P = 6 × 0.5 = 3 W
Ammeter in series, voltmeter in parallel. An ammeter has very low resistance; a voltmeter has very high resistance.
✏️ Quick check · 1 of 7
Current unit
The unit of electric current is the:
Named after Ampère.
✏️ Quick check · 2 of 7
Parallel voltage
In a parallel circuit, each branch has:
Same two points.
✏️ Quick check · 3 of 7
Series current
In a series circuit, the current:
One path.
✏️ Quick check · 4 of 7
Meters
An ammeter must be connected:
It measures the flow.
✍️ Type your answer · 5 of 7
A charge of 6 C flows past a point in 3 s.
What is the current, in A?
Enter an exact answer, without rounding.
A
Charge ÷ time.
✍️ Type your answer · 6 of 7
A 5 Ω resistor carries a current of 2 A.
What is the voltage across it, in V?
Enter an exact answer, without rounding.
V
V = IR.
✍️ Type your answer · 7 of 7
A 3 Ω and a 6 Ω resistor are in series.
What is the total resistance, in Ω?
Enter an exact answer, without rounding.
Ω
Add them.
📝 Lesson quiz · 5 questions
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Re-read the lesson, then give it another go.
Question 1 of 5
A 12 V supply is across a 4 Ω resistor. The current is:
Question 2 of 5
A lamp has 6 V across it and 0.5 A through it. Its power is:
Question 3 of 5
Two identical 10 Ω resistors are in parallel. The total resistance is:
Question 4 of 5
Adding more resistors in series makes the current:
Question 5 of 5
Current in a metal wire is a flow of:
Final Challenge
🏆 12-Question Challenge Quiz
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
0 / 12
💪 Keep practicing — go back through the lessons and retake the quiz!
Q1 · Vectors & Scalars
Which statement about vectors is correct?
Q2 · Vectors & Scalars
After a full lap of a circular track, a runner’s displacement is:
Q3 · Equations of Motion
A car moving at 15 m/s brakes at −3 m/s². How long does it take to stop?
Q4 · Equations of Motion
A car moving at 12 m/s accelerates at 2 m/s² for 4 s. Its final speed is:
Q5 · Free Fall & Projectiles
A ball is thrown straight up at 20 m/s. Its speed after 1 s is:
Q6 · Free Fall & Projectiles
For a projectile, which quantity links the horizontal and vertical motion?
Q7 · Friction & Springs
A spring’s extension doubles. The force it exerts:
Q8 · Friction & Springs
Friction usually transfers kinetic energy to:
Q9 · Energy Conservation
A roller coaster car has 6000 J of E_p at the top. With no friction, its E_k at the bottom is:
Q10 · Energy Conservation
As a skydiver falls, gravitational potential energy mainly becomes:
Q11 · Current, Voltage & Resistance
A 9 V battery drives 3 A through a resistor. The resistance is: