Nuffield Mathematics teaching resources are for use in secondary and further education

# FSMQ Level 3 (pilot) Dynamics scheme of work

Before starting this Advanced (Level 3) FSMQ students are expected to have acquired the skills and knowledge associated with a Functional Mathematics course at Level 2, or equivalent.

Candidates will also need knowledge of the following.

• Trigonometry: Use of sin, cos and tan (but not the Sine or Cosine rules)
• Algebra: Collection of like terms and solution of linear equations such as 3 + 5t =  24  – 5t
Solution of a quadratic equation by at least one of the following methods: use of graphic calculator, use of the formula (which must be memorised):   $\frac{-&space;b~\pm&space;\sqrt{b^2&space;-&space;4ac}}{2a}$, Factorisation (where appropriate), completion of the square.

A suggested work scheme showing topics and methods to be covered is given below, but the order and time allocations can be varied to suit different groups of students.

Throughout the course, students should learn how to use mathematical models to solve problems, making assumptions to create a simple model of a real situation.  Experimental or investigational methods should be used to explore how the mathematical model relates to the actual situation.  Modelling will include the appreciation that it is appropriate at times to treat relatively large moving bodies as point masses, that the friction law F =µR is experimental, and that the force of gravity can be assumed to be constant only under certain circumstances. The terms: light, smooth, rough, inextensible, thin and uniform should be introduced as soon as possible and used wherever relevant.  Students should be encouraged to comment on the modelling assumptions made, including occasions when terms such as particle, light, inextensible string, smooth surface, and motion under gravity are used.

 Topic area Content Nuffield resource The links below go to pages from which you can download the resources, some recently revised. Sketching and interpreting kinematics graphs (4 hours) Find out what students already know about displacement, speed, velocity and acceleration, and discuss the difference between vector and scalar quantities. Use gradients and area under graphs to solve problems (N.B. Calculus NOT required). Find average speed and average velocity. Model the motion  Six sets of cards for learners to match, each set giving a scenario, displacement-time graph and velocity-time graph. Slide presentation includes all the graphs (animated to aid class discussion). Motion in one dimension with constant acceleration (6 hours) Use graphs to derive the equations $v&space;=&space;u&space;+&space;at$, $s&space;=&space;ut&space;+&space;\frac{1}{2}&space;at^2$ , $s&space;=&space;\frac{1}{2}&space;\left&space;(&space;u&space;+&space;v&space;\right&space;)t$ ,  $v^2&space;=&space;u^2&space;+&space;2as$ Learn these formulae and use them to solve problems involving the position, velocity, speed and acceleration of a particle moving in one dimension with constant acceleration. Constant acceleration equations   Worksheet leading learners through the derivation of these equations (with matching Presentation) Vertical motion under gravity (5 hours) Carry out an experiment to verify Galileo's theory regarding vertical motion. Apply the constant acceleration formulae to vertical motion under gravity with the acceleration due to gravity taken as 9.8 ms–2. Falling ball   Introduces Galileo's claim, shows how to set up a model and gives instructions for an experiment to validate the theory. Runaway train   An activity in which students model the motion of a train rolling down a slope with Powerpoint presentation to aid class discussion.

 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Forces (3 hours) Introduce forces. Make an elastoscale and use it to measure and investigate forces. Draw force diagrams (identifying those forces present) and clearly label diagrams, distinguishing between forces and other quantities such as velocity. Include force of gravity W = mg with g = 9.8 (Newton’s Universal Law not required), tensions in strings and rods and resistance. Measure forces   Introduction to forces - includes instructions on how to make and use an elastoscale to measure and investigate forces. Force diagrams   Learners show forces acting on a variety of objects in real situations. Newton's Laws (8 hours) Introduce Newton’s three laws of motion (for a particle of constant mass).  Use knowledge that the resultant force is zero to find unknown forces on bodies which are at rest or moving with constant velocity. (Not resolution of forces or components of forces.) Apply Newton’s second law in the form F = ma to particles moving with constant acceleration. Include finding the acceleration of a body, if the forces acting are specified, or unknown forces if the acceleration is given. Newtonian modelling  Introduces Newton's Laws and provides practice in applying them in a wide range of contexts. Friction  (5 hours) Carry out an experiment to investigate friction. Discuss limiting friction, the coefficient of friction and the relationship F =μR Use F =μR as a model for dynamic friction to solve problems involving motion on a rough surface. Investigating friction   Activities to investigate friction leading to summary of main points. Solve friction problems   Problems to solve in real contexts. Projectiles  (8 hours) Use the equations $x&space;=&space;V~\textup{cos}~\alpha~t$  and $y&space;=&space;V~\textup{sin}~\alpha~t&space;-&space;\frac{1}{2}&space;gt^2$   to solve problems involving motion of an object under uniform gravity in a vertical plane. Include finding initial speed and/or the angle of projection and modification of the equations to take account of the height of release. (Note that students will be expected to state the equations and be aware of the assumptions made.) Calculate range, time of flight and maximum height.  (Note that formulae for the range, time of flight and maximum height should not be quoted in AQA examinations. Also inclined plane and problems involving resistance will not be set by AQA and use of the identity $2\theta&space;=&space;2~\textup{sin&space;}~\theta&space;~\textup{cos}~\theta$ will not be required.) Galileo's projectile model   Introduces Galileo's model and includes an experiment that learners can carry out to validate it. The presentation can be used to introduce the activity and aid class discussion about the results. Projectile problems   Includes worked examples of typical projectile problems and problems set in real contexts for learners to try.

 Topic area Content Nuffield resources The links below go to pages from which you can download the resources, some recently revised. Momentum (5 hours) Introduce the concept of momentum (momentum = mv). Apply the principle of conservation of momentum to two particles for direct impacts in one dimension.  (Note that knowledge of Newton's law of restitution is not required  by AQA.) Use Force = rate of change of momentum Collisions   Introduces and provides practice in using the principle of conservation of linear momentum. Vectors (8 hours) Understanding of a vector; its magnitude and direction. Add and subtract vectors, multiply a vector by a scalar. Magnitude and direction of quantities represented by a vector.                                 Solve problems such as finding the time when a particle is at a specified position or has a specified velocity, or finding the position, velocity or acceleration of a particle at a specified time. Use the constant acceleration equations in vector form i.e. using  $\textbf{v}&space;=&space;\textbf{u}&space;+&space;\textbf{a}t$ ,$\textbf{s}&space;=&space;\textbf{u}t&space;+&space;\frac{1}{2}\textbf{a}t^2$, $\textbf{s}&space;=&space;\frac{1}{2}\left&space;(&space;\textbf{u}&space;+&space;\textbf{v}&space;\right&space;)t$  to find the position, velocity, speed and/or acceleration of a particle moving in two dimensions with constant acceleration.   (Note that AQA examinations questions will be set using the column vector notation, and resolution of velocities will not be required.) Use momentum as a vector in two dimensions. Apply Newton’s three laws of motion in two dimensions using vectors. Vectors   Information sheet and examples of vectors used in real life contexts (also included in presentation). Eight varied problems for students to solve. Revision (8 hours) Revise topics. Work through revision questions and practice papers. Discuss the data sheet - make up and work through questions based on it.

Page last updated on 25 January 2012