Vectors and Scalars Explained

• A vector possesses both magnitude and direction, while a scalar has only magnitude.
• Scalars are generally non-negative, whereas vectors can be negative, indicating a direction.

Examples of Vectors

• Velocity
• Acceleration
• Force
• Momentum
• Displacement

Examples of Scalars

• Speed

• Distance

• Time

• Energy

• Mass

• Displacement becomes zero at the height of a cliff, positive above it, and negative below it when a ball is thrown from the cliff.

• The "0" reference point for a vector is arbitrary and can be assigned as needed.

• Speed turns into velocity when a direction is specified. As an example 10ms⁻¹ becomes a speed when direction is not considered and becomes a velocity when direction is at 30°.

• An object moving at a constant speed on a roundabout has constant speed but variable directional velocity, this implies acceleration.

• Vectors may be represented graphically using arrows, where the arrow's size corresponds to the vector's magnitude.

Object Interaction

• Forces arise when objects interact, either through pushing or pulling, and can be categorized as:

Non-Contact Forces

• Act when objects are physically separated.
• Electrostatic forces are caused by charges and can be attractive or repulsive.
• The mass of objects causes gravitational attraction.

Contact Forces

• Act when objects are physically touching.
• Normal contact force is felt in the opposite direction to contact. The force acts normal to the planes of contact.
• Friction is surface roughness causing resistance when objects are in contact and motion.

Gravity

• All matter exhibits a gravitational field, leading to mutual attraction.
• Larger masses result in stronger gravitational fields and greater attraction forces.

Weight

• Weight is the force on a mass due to gravity (measured in Newtons)
• Weight is calculated as weight = mass × gravitational field strength, or W = mg = m × 10.
• A force meter, also called a calibrated spring-balance, is used to measure weight
• Weighing scales measure force exerted and divide by 10 to estimate mass.
• On Earth, the gravitational field strength (g) is approximately 9.8.
• Mass remains constant regardless of location, but weight varies with gravitational field strength.
• Objects in free fall accelerate at g, which is approximately 10ms⁻².
• The weight of an object is considered to act at its center of mass.

Resultant Force

• The resultant force consolidates all forces acting on an object.
• Resultant Force = sum of forces acting in same direction- sum of forces acting in opposite directions.

Skydiver example

• Air resistance and weight are the forces that act on a skydiver
• Initially, sky diver has no air resistance, resultant force acting on them is weight, so accelerates
• As skydiver falls, air resistance increases with speed, and the resultant force decreases= weight-(air resistance). Acceleration decreases
• Eventually, air resistance equals weight, resultant force is zero, so no acceleration occurs and terminal velocity is reached

Resolving Forces

• A force (F) acting at an angle (θ) can be divided into parallel (Fcosθ) and perpendicular (Fsinθ) components relative to the ground.
• Components can be calculated using Pythagoras' Rule, as expressed by the formula: F² = (Fcosθ)² + (Fsinθ)².

Work

• Work Done = Force × Distance. (W = Fs)
• Work Done (W) is measured in joules (J), Force (F) is measured in newtons (N), and Distance (s) is measured in meters (m).
• Distance is defined as distance moved along the line of action of the force.
• Work is done by force when energy is transferred.
• When a book is lifted 1m vertically, work is done against gravity. Work is not done to the book if it is moved 2m to the right as there is no force acting to it.
• Lifting a book transfers energy from muscles, increasing gravitational potential energy.
• 1 joule = 1 newton-metre
• Work done against frictional forces causes a rise in temperature of the object.

Springs

• Applying more than one force is necessary to stretch, bend, or compress an object.
• Applying a single force will cause movement.
• Pulling an object in opposite directions on either side will result in stretching.
• Fixing an object at one point and stretching it applies force to it.
• Deformation means changing shape.

Elastic Deformation

• Object returns to its original shape when forces are removed.
• An elastic band is an example.

Plastic Deformation

• Object does not return to its original shape after forces are removed.
• A pulled spring that has been deformed is an example.

Hooke's Law

• The extension of a spring is directly proportional to the force applied, as long as the limit of proportionality is not exceeded.
• F = kx.
• The spring constant (k) is measured in Nm-1
• Extension(x) is measured in m
• A Force/Extension graph's linear section represents elastic behavior, following Hooke's Law with a gradient of k.
• The point at which the graph stops being linear is the limit of proportionality.
• The non-linear section signifies plastic behavior, not following Hooke's Law.
  • Shallow gradients mean lots of extension for a small amount of force, and easy stretching.
• The point at which the metal snaps shows a brittle material.

Work Done

• Work Done = 1/2 kx².
• When stretching or compresses a spring, elastic potential energy is stored.
• Provided the spring does not deform inelastically: The work done on the spring equals the elastic potential energy stored.

Moments and Rotation (Physics only)

• For an object attached to a pivot point:
  • A force applied along a line passing through the pivot will not cause rotation, instead keeps object still.
  • A force applied with a distance between pivot and line of action rotates the object.
• If a Force is applied non-perpendicularly, consider perpendicular distance from pivot to line of force

Moment of a Force

• Moment of a Force = force × perpendicular distance
• M = Fd
• Moment of a force (M) is measured in newton-metres (Nm), force (F) is measured in newtons (N), and distance (d) is measured in metres (m).
• In equilibrium: sum of anticlockwise moments = sum of clockwise moments.

Levers and Gears (Physics only)

• Gears can change speed, force, and/or rotation direction.

First Gear Supplying the Force

• Smaller gears (fewer teeth): the second gear turns faster with less force, and turns in the opposite direction.
• Larger gears (more teeth): the second gear turns slower with more force, and turns in the opposite direction.
• To increase power, larger gears are used.

Pressure (Physics only)

• Gas particles move randomly exert forces on container, hence pressure.
• Equation: pressure, p = force / area = F/A
• Pressure (p) is measured in pascals (Pa), force (F) is measured in newtons (N), and area (A) is measured in metres squared (m²).
• Pressure produces a net force at right angles to any surface.

Pressure in a Fluid (Physics only)

• An object floats if its weight is less than the weight of the water it displaces.
• A 1000kg boat sinks until it has displaced 1000kg of water, unless the boat fully submerges.
• Pressure in a liquid varies with depth and density, resulting in force on submerged object.
• The buoyancy force counteracts the object's weight and is equal to the weight of the fluid displaced.
• A object floats if its density is lower than water, hence weight of displaced water is greater than object.
• Greater depth results in greater pressure, thus greater force is felt.
• Pressure due to a column of liquid = height of column × density of liquid × g.
• p = hpg
• Pressure (p) is measured in pascals (Pa), height (h) is measured in meters (m), density (p) is measured in kilograms per metre cubed (kg/m³), and gravitational field strength (g) is normally 10 (N/kg).

Upthrust

• An object experiences greater pressure on the bottom surface, a resultant force upwards is created: upthrust.

• Earth's Atmosphere: a thin layer of less dense air with increasing altitude.

• Pressure is the total weight of the air above a unit measure.

• Weight of air is the force of pressure.

• Pressure is lower at a higher elevation because there are less air molecules.

Idealised Assumptions, for a simple model of the atmosphere

• Isothermal, therefore all at the same temperature.
• Transparent to solar radiation.
• Opaque to terrestrial radiation.

Force and Motion

• Distance is an objects movement, irrespective of direction (scalar)

• Displacement includes distance, straight line from start to finish and direction (vector)

• Speed is a scalar

• Velocity is a vector and specified with direction

• Constant Circular motion means constant velocity, so there must be acceleration.

• The speed of a moving object is rarely constant.

• Typical Speeds: (ms-1)

  • Walk 1.5
  • Run 3

• Depending on lengths involved, use appropriate units for distance measured in mm, cm, m and km and time measured in ms, s, mins and hours.

• Speed = distance/time

• Average speed for non-uniform motion:

  • total distance/ total time
  • time = distance/ speed

Graphs

• Displacement - time graphs measures Gradient is the value for velocity.

  • A sharper line signifies faster speed.
  • Negative gradient indicates returning to the start -Horizontal line means stationary.
  • 0 Distance means that it is back to starting point.
  • Curved Line means that the velocity is changing acceleration.

• Tangent on distance-time graph = speed

• Velocity-Time graphs determine acceleration. Gradient is acceleration. The sharper the gradient, the greater acceleration. Negative gradient equals deceleration

  • Horizontal line means constant speed
  • 0 Velocity means that it is stationary.
  • Area under line equals distance travelled.

• Falling in a fluid shows objects will initially fall under gravity (9.8 m/s²), but the object will move at terminal velocity

Equations

• Average Speed = Total Distance/Total Time
• a = (v-u)/ t
• v² = u² + 2as
• Kinetic Energy = 1/2mv²

Newton's First Law

• Object unless acted on by resultant force equals a constant velocity
  • Acceleration equals the velocity change over time . If no resultant force acts, an object remains stationary.
  • When there is no change to uniform velocity , inertia occurs

Newton's Second Law

• The acceleration of an object is proportional to the resultant force acting on the object, and inversely proportional to the mass of the object.
• Force = mass x acceleration
• F = ma
• where F is the force in newtons N, m is the mass in kg and a is the acceleration in m/s².

Inertia

• The measure of how difficult it is to change the velocity of an object
• inertial mass = force/ acceleration = f/a

Newtons Third Law

• Whenever two objects interact, the forces they exert on each other are equal and opposite.
• For a rocket taking off :
  • The rocket exerts a force on the gases being ejected.
  • The gases apply a force equal in magnitude but in opposite direction on the rocket, which lifts it off the surface.
• For a book on a table: The weight of the book (from the Earth) = the pull of the book on the Earth

Vehicle Stopping Distances

• Stopping distance = Thinking + Braking Distances
• Thinking distance (reaction time travelling at speed X meters)
• Braking Distance (time causing the car to slow down at speed Y meters)

Thinking Distance

• Speed
• Affected by reaction time
• Concentration
• Tiredness
• Distractions
• Influence of drugs/alcohol

Braking Distance

• Speed

• Poor road conditions (icy, wet)

• Bald tires (low friction)

• Worn brake pads

• Weight (more passengers)

• Greater the speed, the greater the distance travelled during the same time (reaction time)

• Reaction Times vary 0.2 – 0.9s for each person

• How to measure reaction times by the "ruler drop": Drop a ruler through the person's open hand, the time it takes to catch it can be determined by s = ut + 1/2at² where u = 0 and a = g, so t = √2gs Where s is the distance the ruler travels through the hand

• When a force is applied to the brakes of a vehicle: Work is done by the brakes (by friction) onto the wheel

  • Therefore the vehicle's KE reduces & Wheel temperature increases
  • Greater the speed = greater braking force is needed to stop the car
  • So greater force = greater acceleration causing brakes overheating and a loss of control

Momentum

• momentum = mass × velocity
• p = mv
• Where p is the momentum in kilograms metres per second kgms¯¹, m is the mass in kilograms kg and v the velocity metres per second ms¯¹. Momentum is conserved in ( collisions & explosions) (no external forces are present) momentum is conserved
• total momentum before = total momentum after
• Collisions = momentum is a vector

Changes in Momentum (Physics only)

• Newton's Second Law: Force is equal to momentum(mv-mu)/ time

Safety Features (Physics only)

• When braking hard, there is a large deceleration causing a danger which can cause neck whiplash.

Safety

• Seatbelts strap person in , but also stretch which increases the distance moved slightly & extends the time taken more for passengers. This reduces reduces rate of momentum

Crumple zones

• These reduce acceleration by absorbing energy reduces compacts.

Air Bags

• Airbags provide a safety mechanism to reduce acceleration of head impact or slow down the neck force
• The increase the time taken for the head to stop moving slowing down the force on the neck