Skill 2.A Deriving

How will you recognize when to use this skill?

On a free response question the word derive in the prompt will often be used, but sometimes a promt can ask you to dertermine an expression, which is an invitation to derive, but it means you may not be evaluated on the logical connections as much as the final result. 

Example FR promtps: 

• "Starting with conservation of momentum, derive an equation for v, the speed of the two – block system after the collision, in terms of M_1, M_2, and v_0."

• "Derive an expression for the linear acceleration of the center of mass of the disk. Express your answer in terms of M, θ, R and physical constants as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference booklet."

• "Determine an expression for the angular acceleration of the ball as it falls in terms of I, R, and T and physical constants, as appropriate."

Example MC prompts:

• "Which of the following is a correct an expression for the speed v_A of the block at point A?" where the answer choices are all expressions. 

How to approach a derivation?

First you need to remember that if you put the correct answer and no support you will likely get no points because to derive means to show the origins and logical progression used to develop an answer. To do this well use the following 5 steps: 

• 1. Plan: Select the models you will use and combine to get to the answer.

     • This part is not written out as part of the derivation. You can do it mentally or make notes in an ungraded part of the page. 

     • Start by looking at what variables you are given to work with and your target variable and compare them with the models discussed in class and on the equation table. 

For example, if you have masses and velocities these are used with momentum and kinetic energy together. 

• 1. • Think about the big ideas and assumptions needed to use those models. 

For example, conservation of momentum applies well to collisions, but conservation of energy is often difficult to track for a collision, so you will select conservation of momentum over conservation of energy every time there is a collision. 

• 1. • Think about what variables the model you selected have that are not provided and find equations that will allow you to substitute those out and replace them with variables you do have. 

For example, if you need to relate the mass, velocity, and force with a distance traveled there are no equations that link those 3, but you can consider starting with a = ΣF/m and v^2=v^2 + 2ad. 

• 2. Label a general equation from the AP Equation Table, physics law, or definition as Equation 1

     • Example of a general principal:
       Equation 1) Conservation of linear momentum states that for isolated collisions total momentum before the collision equals total momentum after the collision. 

     • Example of starting with an equation from the chart:
       Equation 1) a_c = v^2/r

  3. Substitute in any zeroes and variables given in the problem (and constants)

     • Substitute directly below the equation you just referenced so it has a logical mathematical progression.  

     • If any values are zero, you should simplify the equations as soon as possible to make them easier to work with. Write the briefest explanation of why you are are inserting a zero off to the side. 

     • Make sure that you use your capitals where they use capitals and subscripts used in the problem as indicated. 

     • Do not substitute 9.8 or 10 in for g. It is a constant that is allowed, and preferred, in expressions and it limits your work to only function at the surface of the earth. Factors, like 1/2 or 2, are also allowed as a natural progression of using terms in an equation or conditions of the scenario.

  4. Repeatedly substitute out variables you are not allowed

     • Check your expression to identify any variables not in the allowed list and then find equations that match your scenario (hopefully identified in step 1) that you can use to substitute those out. 

     • List any new equations off to the side of the main progression, rearrange and simplify them (step 3) as needed and number them (equation 2, equation 3, etc). Then either describe with a few words or indicate with arrows to show where you are substituting them.   

     • Keep checking and substituting until you have an expression with only allowed terms and constants (like π, G, and g). You do not have to use all the allowed variables. Sometimes there will be an extra one or two unused in the end. 

  5. Simplify your expression if needed and solve for the variable asked for. 

     • On free response questions there are generally not any points that will require simplification, but on multiple choice questions the simiplified versions will be shown, so you should practice simplifying so you recognize the right answers. 

     • If the question asks to "write an equation relating [two variables]" you do not need to solve it for either one, but if it says "derrive an expression for the final velocity" then your answer should be in the form of v = . . .             

*Simplifying is often a step students make mistakes on and there are 2 approaches to reducing this problem 1) you can simplify early when the equations are less complex (could have been done after the step where eq 3 was substituted in), or 2) Wait until the end so at least the unsimplified expression is correct. If you have time you can try it both ways and cross out the way you are less comfortable with. DO NOT leave 2 methods of deriving on your page as we are instructed to grade the less correct version. 

Notice that M is not part of the final expression, I did not repalce G with 6.67x10^-11, and that the main line of reasoning is from top to bottom, or to the right, at the end, in a natural mathematical flow. 

Note that there were a lot of steps in this derivation, which means there will be several partial points you can earn along the way. Make sure not to leave these blank! You will often get points of combining 2 equations several times, so if you can see any points of connection between the variables that seem to apply to the scenario, you should at least put them.

How to get better at derivations:

1. Practice! Practice! Practice! You should absolutely be practicing this with every problem we solve in class or online even if you are given numerical values. Just derive the expression first and plug in the values at the end and you will build confidence and speed at doing these. 

2. Sometimes the hardest part is knowing where to start. When you are trying multiple choice questions identify which main law, definition, or mathematical relationship should be used in a scenario based solely on question stems (scenario description), without looking at the question or the response choices. 

3. Check the Algebra Reminders and Common Errors page to make sure you are not making simple algebra mistakes. 

4. Sometimes being flexible is the hardest part. We may think all angled ramps are asking for an acceleration, as we did in example 1, but with tiny variations in info provided or asked about angled ramps can be used in kinematics, energy, momentum, oscillation, and rotational situations. With each  questino you practice with (especially on progress checks), ask yourselfe what other questions they could ask with the same question stem that would connect or start with a different model. What other info would you need, if any?

This review packet was made by a fellow teacher and is a great place to review derivations. You will not be able to do all of them until we have finished learning through unit 7, but you can try them. My students can find the solutions in Google Classroom.