Scalars, Vectors & Adding Forces

Think of scalars as simple measurements - they're just numbers with units like distance (5m) or speed (30 mph). Vectors are trickier because they have both size and direction, like displacement, force, and velocity.

When adding vectors that are perpendicular (at 90°), you'll use Pythagoras' theorem. For example, if forces of 5N and 12N act at right angles, the resultant force is √$5² + 12²$ = 13N. Use trigonometry (tan θ = opposite/adjacent) to find the direction.

For forces that aren't at right angles, you'll need either scale drawings or the cosine rule: a² = b² + c² - 2bc cos θ. This method works for any angle between the forces.

Quick Tip: Always sketch the vectors first - it'll help you visualise the problem and avoid mistakes with directions.

Resolving Vectors & Equilibrium

Resolving vectors means splitting them into horizontal $x$ and vertical $y$ components using trigonometry. Remember: horizontal component = V cos θ and vertical component = V sin θ.

For a 10 m/s velocity at 30°, the horizontal component is 10 × cos 30° = 8.7 m/s, and the vertical component is 10 × sin 30° = 5 m/s. This technique is essential for analysing forces on slopes or projectile motion.

When objects are in equilibrium, all forces balance out perfectly. This means the sum of all horizontal components equals zero, and the sum of all vertical components equals zero. You can prove equilibrium by showing these conditions are met.

Moments are forces that cause rotation, calculated as Force × perpendicular distance. For equilibrium, clockwise moments must equal anticlockwise moments.

Remember: The centre of mass is where an object's weight appears to act - it's at the geometric centre for uniform objects.

Forces on Slopes & SUVAT Equations

When dealing with forces on slopes, always resolve the weight into components parallel and perpendicular to the slope. For a 50N force at 15° to the horizontal: parallel component = 50 sin 15° = 12.9N, perpendicular component = 50 cos 15° = 48.3N.

The SUVAT equations are your best friends for motion problems. They connect displacement $s$, initial velocity $u$, final velocity $v$, acceleration $a$, and time $t$. Choose the equation that contains three known values and one unknown.

For a stone dropped from 50m: using v² = u² + 2as gives v = √(2 × 9.81 × 50) = 31.3 m/s. Then use v = u + at to find the time: t = $31.3 - 0$/9.81 = 3.2 seconds.

Pro Tip: Always list your known values (s, u, v, a, t) before choosing which SUVAT equation to use - it'll save you time in exams.

Advanced Vector Calculations

Adding vectors gets more complex when they're not perpendicular. You'll need the cosine rule for finding resultants and the sine rule for directions. These are the same rules from trigonometry, just applied to force problems.

For forces of 3.0N and 5.0N with 120° between them, the resultant is F = √$3² + 5² - 2×3×5×cos120°$ = 7.0N. The sine rule then gives you the direction.

Vector resolution offers an alternative approach. Break each vector into horizontal and vertical components, add these components separately, then combine using Pythagoras to find the resultant magnitude and direction.

Study Smart: Practice both methods - cosine/sine rules and vector resolution. Some problems are easier with one method than the other.

Vector Resolution Method

This method is often cleaner for complex vector problems. Break each force into horizontal and vertical components, then add all horizontal components together and all vertical components together.

For a 5.0N force at 60° plus a 3.0N horizontal force: vertical total = 5.0×sin60° = 4.33N, horizontal total = 5.0×cos60° + 3.0 = 5.5N. The resultant is √$5.5² + 4.33²$ = 7.0N.

Use SOH CAH TOA to find the direction: tan θ = 4.33/5.50, so θ = 38° from horizontal. This systematic approach works for any number of forces.

Exam Hack: Vector resolution is usually faster for problems with more than two forces - it's worth mastering this method properly.

Projectile Motion & Terminal Velocity

Projectile motion treats horizontal and vertical components completely independently. For a projectile launched at 20 m/s at 60°: horizontal component = 20×cos60° = 10 m/s (stays constant), vertical component = 20×sin60° = 17.3 m/s (changes due to gravity).

Maximum height occurs when vertical velocity becomes zero. Using v² = u² + 2as with v = 0, u = 17.3 m/s, and a = -9.81 m/s²: maximum height = 15.3m.

Terminal velocity happens when air resistance equals the driving force. A skydiver initially accelerates because weight exceeds air resistance, but as speed increases, air resistance grows until forces balance.

Key Point: Air resistance affects both horizontal and vertical motion, reducing both maximum height and range compared to motion in a vacuum.

Newton's Laws & Momentum

Newton's three laws are fundamental: (1) objects continue at constant velocity unless acted upon by a resultant force, (2) F = ma, and (3) every action has an equal and opposite reaction.

Momentum $p = mv$ is conserved in collisions - momentum before equals momentum after. The impulse-momentum theorem connects force and time: F∆t = ∆mv, meaning a larger impact time reduces the force.

This explains why crumple zones, seatbelts, and airbags work - they increase collision time, dramatically reducing the force on passengers during crashes.

Real-world Connection: Understanding momentum conservation helps explain everything from car crashes to rocket propulsion - it's physics you can see everywhere.

Collisions, Work & Power

Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions only conserve momentum - kinetic energy is lost as heat, sound, or deformation. If objects stick together after collision, it's definitely inelastic.

Work is force times distance in the direction of motion: W = Fs cos θ. When force varies, work equals the area under a force-displacement graph.

Power measures the rate of energy transfer: P = W/t = Fv. This relationship shows why car engines need more power at higher speeds to maintain acceleration.

Formula Focus: Remember P = Fv - this connects power, force, and velocity in a way that's incredibly useful for vehicle dynamics problems.

Conservation of Energy

Energy cannot be created or destroyed, only transferred between different forms. In a closed system, total energy remains constant - this is the principle of conservation of energy.

The classic example is throwing a ball upward: kinetic energy converts to gravitational potential energy as it rises, stops momentarily when all KE becomes PE, then converts back to KE as it falls.

Use the relationship mgh = ½mv² to solve energy problems. If a 0.05kg ball is dropped 0.1m: PE = 0.05×9.81×0.1 = 0.049J, giving final velocity v = √$2×0.049/0.05$ = 1.4 m/s.

Problem-solving Tip: Energy conservation problems are often easier than force-based approaches - look for opportunities to use PE = KE relationships.

Material Properties & Hooke's Law

Hooke's Law states that extension is proportional to applied force: F = k∆L, but only up to the limit of proportionality. Beyond the elastic limit, materials deform permanently.

Elastic strain energy stored in stretched materials equals ½F∆L - this is the area under a force-extension graph. This energy is recoverable if the material returns to its original shape.

Materials behave differently under stress: brittle materials snap with little extension, plastic materials deform permanently, and elastic materials return to original shape when force is removed.

Safety Application: Crumple zones in cars are designed to deform plastically, absorbing kinetic energy and protecting passengers during collisions.