IB Physics # Topic 2: Mechanics

## 2.1 Motion

Essential idea:

Motion may be described and analysed by the use of graphs and equations.

Nature of science:

Observations: The ideas of motion are fundamental to many areas of physics, providing a link to the consideration of forces and their implication. The kinematic equations for uniform acceleration were developed through careful observations of the natural world.

Understandings:

Applications and skills

• Determining instantaneous and average values for velocity, speed and acceleration
• Solving problems using equations of motion for uniform acceleration
• Sketching and interpreting motion graphs
• Determining the acceleration of free-fall experimentally
• Analysing projectile motion, including the resolution of vertical and horizontal components of acceleration, velocity and displacement
• Qualitatively describing the effect of fluid resistance on falling objects or projectiles, including reaching terminal speed

Guidance

• Calculations will be restricted to those neglecting air resistance
• Projectile motion will only involve problems using a constant value of g close to the surface of the Earth
• The equation of the path of a projectile will not be required

Data Booklet reference

• v = u + at
• s = ut + (1/2)at2
• v2 = u2 + 2as
• s = (1/2) (v + u) / t
• The v represents the final velocity. The u represents the initial velocity. The t represents the elapsed time. The a represents the constant acceleration. The s represents the displacement. Note that many textbooks use the symbol "Δx" for s and v0 for u.

International-mindedness

• International cooperation is needed for tracking shipping, land-based transport, aircraft and objects in space.

Theory of knowledge

• The independence of horizontal and vertical motion in projectile motion seems to be counter-intuitive. How do scientists work around their intuitions? How do scientists make use of their intuitions?

Utilization

• Diving, parachuting and similar activities where fluid resistance affects motion
• The accurate use of ballistics requires careful analysis
• Biomechanics (see Sports, exercise and health science SL sub-topic 4.3)
• Quadratic functions (see Mathematics HL sub-topic 2.6; Mathematics SL sub-topic 2.4; Mathematical studies SL sub-topic 6.3)
• The kinematic equations are treated in calculus form in Mathematics HL sub-topic 6.6 and Mathematics SL sub-topic 6.6

Aims

• Aim 2: much of the development of classical physics has been built on the advances in kinematics
• Aim 6: experiments, including use of data logging, could include (but are not limited to): determination of g, estimating speed using travel timetables, analysing projectile motion, and investigating motion through a fluid
• Aim 7: technology has allowed for more accurate and precise measurements of motion, including video analysis of real-life projectiles and modelling/simulations of terminal velocity

## 2.2 Forces

Essential idea:

Classical physics requires a force to change a state of motion, as suggested by Newton in his laws of motion.

Nature of science

(1) Using mathematics: Isaac Newton provided the basis for much of our understanding of forces and motion by formalizing the previous work of scientists through the application of mathematics by inventing calculus to assist with this. (2) Intuition: The tale of the falling apple describes simply one of the many flashes of intuition that went into the publication of Philosophiæ Naturalis Principia Mathematica in 1687.

Understandings

Applications and skills

• Representing forces as vectors
• Sketching and interpreting free-body diagrams
• Describing the consequences of Newton’s first law for translational equilibrium
• Using Newton’s second law quantitatively and qualitatively
• Identifying force pairs in the context of Newton’s third law
• Solving problems involving forces and determining resultant force
• Describing solid friction (static and dynamic) by coefficients of friction

Guidance

• Students should label forces using commonly accepted names or symbols (for example: weight or force of gravity or mg)
• Free-body diagrams should show scaled vector lengths acting from the point of application
• Examples and questions will be limited to constant mass
• mg should be identified as weight
• Calculations relating to the determination of resultant forces will be restricted to one- and two-dimensional situations

Data Booklet reference

• F = ma
• Ffµs R
• Ff = µd R
• The F represents the net force acting on an object, and F = ma is often written ΣF = ma because the net force is the sum of all forces acting on an object. The m represents the mass. The a reperesents the acceleration. The Ff represents the frictional force. The R represents the reaction force (or normal force). The µsrepresents the coefficient of static (non-sliding) friction. The µd represents the coefficient of dynamic (sliding) friction. Some textbooks use N (the normal force) for R and µk (coefficient of kinetic friction) for µd. Note that F = ma works only if the mass of the system remains constant.

Theory of knowledge

• Classical physics believed that the whole of the future of the universe could be predicted from knowledge of the present state. To what extent can knowledge of the present give us knowledge of the future?

Utilization

• Motion of charged particles in fields (see Physics sub-topics 5.4, 6.1, 11.1, 12.2)
• Application of friction in circular motion (see Physics sub-topic 6.1)
• Construction (considering ancient and modern approaches to safety, longevity and consideration of local weather and geological influences)
• Biomechanics (see Sports, exercise and health science SL sub-topic 4.3)

Aims

• Aim 2 and 3: Newton’s work is often described by the quote from a letter he wrote to his rival, Robert Hooke, 11 years before the publication of Philosophiæ Naturalis Principia Mathematica, which states: “What Descartes did was a good step. You have added much several ways, and especially in taking the colours of thin plates into philosophical consideration. If I have seen a little further it is by standing on the shoulders of Giants.” It should be remembered that this quote is also inspired, this time by writers who had been using versions of it for at least 500 years before Newton’s time.
• Aim 6: experiments could include (but are not limited to): verification of Newton’s second law; investigating forces in equilibrium; determination of the effects of friction

## 2.3 Work, energy, and power

Essential idea:

The fundamental concept of energy lays the basis upon which much of science is built.

Nature of science

Theories: Many phenomena can be fundamentally understood through application of the theory of conservation of energy. Over time, scientists have utilized this theory both to explain natural phenomena and, more importantly, to predict the outcome of previously unknown interactions. The concept of energy has evolved as a result of recognition of the relationship between mass and energy.

Understandings

• Kinetic energy Definition and test
• Gravitational potential energy Definition and test
• Elastic potential energy Mass-spring lab
• Work done as energy transfer The centrifuge brain project
• Power as rate of energy transfer Definition and test
• Principle of conservation of energy Skate park basics Skate park The 10 biggest jumps ever Nitro Circus
• Efficiency Sample Sankey diagrams

Applications and skills

• Discussing the conservation of total energy within energy transformations
• Sketching and interpreting force–distance graphs
• Determining work done including cases where a resistive force acts
• Solving problems involving power
• Quantitatively describing efficiency in energy transfers

Guidance

• Cases where the line of action of the force and the displacement are not parallel should be considered
• Examples should include force–distance graphs for variable forces

Data Booklet reference

• W = Fs cosθ
• EK = (1/2)mv2
• EP = (1/2)kΔx2
• ΔEP = mgΔh
• power = Fv
• Efficiency = Wout / Win = Pout / Pin
• The W represents the work done by the force F acting over a displacement s. The θ represents the angle between F and s. The EK represents the kinetic energy of a mass m traveling at a velocity v. The EP represents the elastic potential energy of a spring having a spring constant of k being displaced a distance Δx from equilibrium. The ΔEP represents the change in gravitational potential energy of the mass m during its change in height Δh. The g represents the magnitude of the acceleration due to gravity which is 9.8 ms-2. The power P is the work (or energy change) per unit time.

Theory of knowledge

• To what extent is scientific knowledge based on fundamental concepts such as energy? What happens to scientific knowledge when our under-standing of such fundamental concepts changes or evolves?

Utilization

• Energy is also covered in other group 4 subjects (for example, see: Biology topics 2, 4 and 8; Chemistry topics 5, 15, and C; Sports, exercise and health science topics 3, A.2, C.3 and D.3; Environmental systems and societies topics 1, 2, and 3)
• Energy conversions are essential for electrical energy generation (see Physics topic 5 and sub-topic 8.1)
• Energy changes occurring in simple harmonic motion (see Physics sub-topics 4.1 and 9.1)

Aims

• Aim 6: experiments could include (but are not limited to): relationship of kinetic and gravitational potential energy for a falling mass; power and efficiency of mechanical objects; comparison of different situations involving elastic potential energy
• Aim 8: by linking this sub-topic with topic 8, students should be aware of the importance of efficiency and its impact of conserving the fuel used for energy production

## 2.4 Momentum and impulse

Essential idea:

Conservation of momentum is an example of a law that is never violated.

Nature of science

The concept of momentum and the principle of momentum conservation can be used to analyse and predict the outcome of a wide range of physical interactions, from macroscopic motion to microscopic collisions.

Understandings

• Newton’s second law expressed in terms of rate of change of momentum Concept and movies
• Impulse and force–time graphs Impulse and momentum
• Conservation of linear momentum Conservation of momentum Newton's third law of motion and the NFL
• Elastic collisions, inelastic collisions and explosions Inelastic collision sample and test Explosion sample and test Collisions lab

Applications and skills

• Applying conservation of momentum in simple isolated systems including (but not limited to) collisions, explosions, or water jets
• Using Newton’s second law quantitatively and qualitatively in cases where mass is not constant
• Sketching and interpreting force–time graphs
• Determining impulse in various contexts including (but not limited to) car safety and sports
• Qualitatively and quantitatively comparing situations involving elastic collisions, inelastic collisions and explosions

Guidance

• Students should be aware that F = ma is equivalent to F = Δp / Δt only when mass is constant
• Solving simultaneous equations involving conservation of momentum and energy in collisions will not be required
• Calculations relating to collisions and explosions will be restricted to one-dimensional situations
• A comparison between energy involved in inelastic collisions (in which kinetic energy is not conserved) and the conservation of (total) energy should be made

Data Booklet reference

• p = mv
• F = Δp / Δt
• EK = p2/ ( 2m )
• The p represents the momentum of a mass m traveling at a velocity v. The F represents the net force acting on a mass whose change in momentum Δp occurs over a time interval Δt. The EK represents the kinetic energy of a mass m having a momentum p. For constant mass use F = ma. For changing mass use the momentum form of Newtons 2nd law: F = Δp / Δt.

International-mindedness

• Automobile passive safety standards have been adopted across the globe based on research conducted in many countries

Theory of knowledge

• Do conservation laws restrict or enable further development in physics?

Utilization

• Jet engines and rockets Basic aeronautics
• Martial arts
• Particle theory and collisions (see Physics sub-topic 3.1)

Aims

• Aim 3: conservation laws in science disciplines have played a major role in outlining the limits within which scientific theories are developed
• Aim 6: experiments could include (but are not limited to): analysis of collisions with respect to energy transfer; impulse investigations to determine velocity, force, time, or mass; determination of amount of trans-formed energy in inelastic collisions
• Aim 7: technology has allowed for more accurate and precise measurements of force and momentum, including video analysis of real-life collisions and modelling/simulations of molecular collisions

## Extension Notes for Enrichment

Estes Model Rocketry Unit

for someone wanting to apply all this cool physics to a real (Estes) rocket. There is mathematical modeling, also. I used to use this as my mechanics unit in regular physics years ago. Oh what a joyous independent study for an enterprising student!

Differential calculus

Differential calculus, anyone? After all, Newton invented this stuff for physics. Do you have the right stuff? The PowerPoints (the white icons) are tutorials. The Word documents are practice worksheets. See the additional worksheets below (with keys) for some real calculus.

Visual calculus

This simulation graphs functions and then shows visually both derivatives and integrals (explained below). Calculus grapher

Unit vectors

The boat and the plane equations of relative velocity and plotting airplane courses are the topics of these notes.

Rotational kinematics

IBO only requires you to know the kinematics of linear motion. There is, however, rotational motion other than UCM which has angular displacement, angular velocity, and angular acceleration. These topics are covered in these PowerPoints.

Amusement parks.

Now that you understand rotational kinematics you can understand the next generation amusement park rides. Try the link. The centrifuge brain project

More differential calculus

If you are really a glutton for punishment, feel free to try these additional practice worksheets. Beside each lesson is the associated answer key. Don't cheat, now, boys and girls. The last of this set has no answer key, but it has some really cool and lazy notation for time derivatives.

Integral calculus

Integral calculus, also known as antidifferentiation, is explored in these practice lessons. If you have mastered differentiation it is high time you master how to antidifferentiate. The two go hand in hand.

Center of mass

The center of mass (CM) is that single point in a rotating extended body which follows the simple kinematic equations of motion in response to Newton's second law. The first two PowerPoint lessons explore finding the CM of extended bodies made of discrete point masses. The third lesson explores methods of finding the CM of extended solids which can be constructed from simple symmetric sub-sets of solids. The fourth lesson is how to use integrals to find the CM of more irregular solids and is demonstrated by use of the Riemann sum.

Extended forces and dynamics

This is more college-oriented stuff for you go-getters. Friction is detailed and includes coefficients of friction and details about the drag force and the drag coefficient, Applications of Newton's second law has two-body problems, Conservation of linear momentum shows how to find the friction coefficient, Impulse shows collisions in detail, including 2D collisions, and Elastic and inelastic collisions has elephant-fly collisions.

Statics and fluid dynamics

This set of notes covers the statics of solids and fluids, and fluid dynamics. If you are interested in how buildings and bridges work, if you are interested in the strength of materials, if you are interested in how the hydraulics of your car's braking system works, how submarines and ships work, or how planes fly, look at these extensions.

Rotational work and energy

Of course, if Newton's second law can be generalized to angular motion, so can the concepts of work, energy and power. Here is that lesson.

Newton's law of gravitation

One key example of uniform circular motion is the gravitational force between celestial bodies. This lesson explores the universal law of gravitation as presented by Newton. Gravitation inside the earth explains what would happen if you stepped into a hole drilled through the center of the earth and out to the other side. You might want to come back to this one after mastering Topic 4 and simple harmonic motion. Or, you can wait until we cover Topic 6. Choose your pi-zon.

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