Scalars & Vectors
Scalar vs Vector Quantities
A scalar has magnitude only (e.g. speed, mass, energy, temperature). A vector has both magnitude and direction (e.g. velocity, force, acceleration, displacement).
Resolving Vectors
Any vector can be split into two perpendicular components. For a vector F at angle θ to the horizontal:
Vertical component: Fy = F sinθ
To find the resultant of two perpendicular vectors:
Key Facts
- Vectors are added tip-to-tail; the resultant goes from the start of the first to the end of the last
- For equilibrium, the vector triangle must close (resultant = 0)
- Distance is scalar; displacement is vector
Moments
Principle of Moments
The moment of a force about a point is the force multiplied by the perpendicular distance from the line of action to the pivot.
For an object in equilibrium: the sum of clockwise moments equals the sum of anticlockwise moments about any point.
Couples & Torque
A couple is a pair of equal and opposite forces that produce rotation without translation.
(d = perpendicular distance between the forces)
Exam Tip
When solving moments problems, choose the pivot point wisely — pick the point where an unknown force acts to eliminate it from your equation.
Kinematics (SUVAT)
The Five SUVAT Equations
These equations describe motion with constant (uniform) acceleration in a straight line. The five variables are: s (displacement), u (initial velocity), v (final velocity), a (acceleration), t (time).
2. s = ut + ½at²
3. s = vt − ½at²
4. v² = u² + 2as
5. s = ½(u + v)t
Worked Example
A car accelerates from rest at 2.5 m s−² for 8 s. Find the distance travelled.
Known: u = 0, a = 2.5 m s−², t = 8 s. Find s.
Use s = ut + ½at²
s = 0 + ½ × 2.5 × 8² = ½ × 2.5 × 64 = 80 m
Velocity–Time Graphs
Key features of a velocity–time graph:
- Gradient = acceleration
- Area under the graph = displacement
- A horizontal line means constant velocity (zero acceleration)
- A straight line with positive gradient means constant acceleration
Free Fall & Projectiles
Objects in free fall have acceleration g = 9.81 m s−² downward. For projectile motion, resolve into horizontal (constant velocity) and vertical (SUVAT with a = g) components independently.
Exam Tip
Always list the known SUVAT variables first, then choose the equation that links them to the unknown. Each equation is missing exactly one of the five variables.
Forces & Newton's Laws
Newton's Three Laws of Motion
2nd Law: F = ma (resultant force = mass × acceleration)
3rd Law: If A exerts a force on B, then B exerts an equal and opposite force on A (same type, different objects).
Free Body Diagrams
A free body diagram shows all the forces acting on a single object. Each force is drawn as an arrow from the object, with length proportional to magnitude.
- Weight (W = mg) acts downward from the centre of mass
- Normal contact force (N) acts perpendicular to the surface
- Friction acts along the surface, opposing motion
- Tension acts along a string or rope, away from the object
- Air resistance / drag opposes the direction of motion
Terminal Velocity
When an object falls through a fluid, drag increases with speed. When drag equals weight, the resultant force is zero, acceleration is zero, and the object moves at terminal velocity.
Worked Example
A 75 kg skydiver reaches terminal velocity. What is the air resistance at this point?
At terminal velocity: air resistance = weight
W = mg = 75 × 9.81 = 736 N
Key Facts
- Newton's 3rd law pairs act on different objects and are always the same type of force
- F = ma only applies when mass is constant; the full form is F = Δ(mv)/Δt
- On an inclined plane, resolve weight into components parallel and perpendicular to the slope
Momentum
Conservation of Momentum
Momentum is the product of mass and velocity. It is always conserved in any collision (provided no external forces act).
Total momentum before = Total momentum after
m1u1 + m2u2 = m1v1 + m2v2
Elastic & Inelastic Collisions
Elastic collision: both momentum and kinetic energy are conserved. Objects bounce apart.
Inelastic collision: momentum is conserved but kinetic energy is not (some is converted to heat, sound, deformation). In a perfectly inelastic collision, the objects stick together.
Worked Example
A 2 kg trolley moving at 3 m s−¹ collides with a stationary 4 kg trolley. They stick together. Find the velocity after collision.
m1u1 + m2u2 = (m1 + m2)v
(2 × 3) + (4 × 0) = (2 + 4) × v
6 = 6v ⇒ v = 1 m s−¹
Impulse
Impulse is the change in momentum. It equals the force multiplied by the time for which it acts.
On a force–time graph, the area under the curve equals the impulse.
Exam Tip
Remember that momentum is a vector. Assign a positive direction and use negative values for objects moving the opposite way.
Work, Energy & Power
Work Done
Work is done when a force moves an object through a displacement in the direction of the force.
When θ = 0°, W = Fs. When θ = 90°, W = 0 (force perpendicular to displacement does no work).
Kinetic & Gravitational Potential Energy
The principle of conservation of energy: energy cannot be created or destroyed, only transferred between stores. In a closed system, total energy is constant.
Worked Example
A 0.5 kg ball is dropped from 20 m. Find its speed just before hitting the ground (ignore air resistance).
Ep lost = Ek gained
mgh = ½mv²
v = √(2gh) = √(2 × 9.81 × 20) = √392.4 = 19.8 m s−¹
Power & Efficiency
Efficiency = useful output energytotal input energy × 100%
Key Facts
- 1 watt = 1 joule per second
- At constant velocity, the driving force equals the resistive force, and P = Fv gives the power output
- Efficiency is always less than 100% due to energy dissipation
Materials (Hooke's Law, Young Modulus)
Density & Pressure
Density is a scalar property of a material. It does not depend on the size or shape of the object.
Hooke's Law
Up to the limit of proportionality, extension is directly proportional to the applied force.
where k is the spring constant (N m−¹) and ΔL is the extension.
Stress, Strain & Young Modulus
Strain: ε = ΔLL (no units)
Young Modulus: E = σε = FLAΔL (unit: Pa)
Elastic Potential Energy
This is the area under the force–extension graph (up to the limit of proportionality).
Stress–Strain Graphs
Stress–strain graphs reveal important material properties:
- Limit of proportionality: beyond this, stress is no longer proportional to strain
- Elastic limit: beyond this, the material will not return to its original shape (permanent deformation)
- Yield point: material begins to deform plastically with little increase in stress
- Ultimate tensile stress (UTS): maximum stress the material can withstand
- Fracture point: the material breaks
Elastic & Plastic Deformation
Elastic deformation: the material returns to its original shape when the force is removed. Energy is stored as elastic potential energy.
Plastic deformation: the material is permanently deformed. Energy is dissipated (not recoverable).
Exam Tip
The Young modulus is the gradient of the linear part of a stress–strain graph. Make sure you use stress/strain (not force/extension) when calculating it from a graph.
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