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AP HW: Projectiles 1, numbers 8, 9 & 10

Repost from past year

AP Projectiles #1

Problem 8, 9, 10


Several of you have inquired this evening about questions 8, 9 and 10.  Here are your helps:



In this problem you are given a velocity function which is initially in the x-direction only and an acceleration that is in the negative x-direction as well as the negative y-direction (at least on my version).  In my opinion, the easiest way to do this problem is to break apart the equation into x- and y- directions and do kinematics.  We can do this because a) vectors at right angles are independent (that is to say the y-acceleration does not affect the x-direction of motion) and b) because the acceleration in both directions is constant…note no time dependency in the accel. Equation.


OK, so in the x direction we know

Starting position (origin = 0)

Initial velocity (your value)

Final velocity (0…where it stops moving positively and starts moving negatively)

Acceleration (your value)

Final position (going to solve for this)

Time (can solve for this using a=deltaV/t)


In the y-direction

Starting position (origin = 0)

Initial velocity (0)

Final velocity (Going to be solving for this)

Acceleration (your value)

Final position (going to solve for this)

Time (same as x-direction)



Nothing happening (unless your vector has a k-hat component and I don’t think any should)



Very similar to number 8 except you have a time dependent acceleration in the x-direction, so we will need to use some calculus.

a)      Take derivative, evaluate for t=4s

b)      Set derivative = 0, solve for time

c)       Set velocity (the function you are given) = 0, solve for time.  Do x- and y-directions separately.  Both must be zero at the same time to satisfy question.  If all answers are not real then never. My answer with my numbers is NEVER, but some of you may have a value.

d)      Set v=10, solve for t



Two of you have asked about number 10.  I checked both of your work.  WebAssign is correct here.  You have probably rounded too much.  If you take your answers to 2 or 3 sig figs, rounding only at the end of your solutions, you will probably get it.  Also, be sure you take into account the bullet drop in meters, not centimeters as stated.


AP-C: Linear Motion 3 HW #8, 16, 15

AP C Linear Motion3,

Problem 8

This is a pretty touch problem.  I began working on it this morning and had to stop and try again to find an easier solution using alternative kinematics equations than what I first tried.  The key is recognizing that you can use a system of equations–multiple equations each describing a different aspect of the motion, with multiple unknowns.  If you can find the right set of equations you can end up with a fairly simple solution.  Here’s the one I found that worked best.

To stop a car, you require first a certain reaction time to begin braking. Then the car slows under the constant braking deceleration. Suppose that the total distance moved by your car during these two phases is 56.7 m when its initial speed is 80.0 km/h, and 24.4 m when the initial speed is 47.8 km/h.  These are the numbers from my problem, of course.

(a) What is your reaction time?
(b) Acceleration?

2 parts to each motion:

–Constant velocity before break is applied (during reaction time)
–During acceleration (slowing)

For constant velocity portion: v = x/t
For slowing motion: 2ax = v^2 – v0^2

Now, this is where it gets tricky.  You will use 2ax etc. twice, once for each situation, but, and this is the big key here, you must substitute for x in that equation since the total x given in the problem is BOTH the constant speed distance traveled AND the slowing down distance traveled.  As a result, you will need to subtract the constant velocity distance from the total stopping distance.

Basically you work will look like this:
Scenario #1:
2a(x1-v01*t) = V1^2 – vo1^2

Scenario #2:
2a(x2-v02*t) = V2^2 – vo2^2

X1 & X2 – given distances
V01 & V02 – given enitial velocities
t – reaction time (unknown)
a – acceleration (unknown)
V1 & V2 – final velocities, in this case both = 0

Notice the terms x – V0*t: These are you taking the distance given, and subtracting the pre-braking, constant velocity distance out of this formula, essentially leaving behind just the distance traveled during the negative acceleration.
Now, at this point you have 2 equations with 2 unknowns.  Solve however you wish, but I prefer substitution.  The algebra gets nasty.  I actually had to redo my problem as I made one simple arithmetic mistake…not fun.  Take it step by step.
Have fun!



This is one of my favorite problems on the assignment.  It has elements of number 8 but is much easier.  You know a couple pieces of information.  In my version of the problem I know it falls from height h, and falls 0.49h in the last second.  The easiest way to solve this is to not focus on the last second, but to focus on the first part of the fall instead.  During whatever time that is (one second less than the total fall time) we know it falls 0.51h, the difference between 1 and whatever value of h you were given in your problem.  This lets us set up two equations:

Whole fall: h = (1/2)gt^2
First part: 0.51h = (1/2)g(t-1)^2  <– this is the key.  notice how we used the info from the problem, but we changed it up.

The reason I like to focus on the first part of the fall is because when Vo is zero, you get to cancel that variable.  Makes the algebra MUCH easier.
So, we are left with 2 equations with 2 unknowns.  Solve as you wish.  I generally prefer substitution.  In this case I simply take the first equation and plug in for h in the second, then used my graphing calculator to find the roots of the resulting quadratic to solve for time.  But wait, you say!  It’s quadratic so it has two solutions!  You are absolutely correct, but one of them is impossible.  I will eave it to you to figure out why, but I will say that one solution is outside the domain of the thing you are solving for.


Most common problem in dealing with this question is that the time given in your problem in part A–50ms–is 0.050s, not 0.50s.  Remember the things on the stopwatch everyone calls milliseconds are in fact NOT MILLISECONDS.  This is why a few of you have gotten answers with are simply 10x too big.  Solving this question then is a simple application of a standard displacement function for constant acceleration, that kinematics equation that you know and love:

x=Vot + .5at^2, of course Vo = ?

Also note the units on the answer.

As for the rest of the problem you can either calculate each one individually, or recognize that because of the constant acceleration of gravity you are dealing with a quadratic relationship between distance fallen and time.  That is to say, in two time the amount of time, the object will fall 4 times as far, or in three times the time, nine times as far, etc.  Work it out if you are confused and look for a quadratic pattern in your answers.


AP-1: Dimensional analysis practice and upcoming quiz notes

Solutions to some of the practice exercises from your notes:

AP-1- Symbolic alg and dimensional analysis practice

Relevant power point notes from class:

Metric system and conversions…you will remember this one from the summer packet if you used it.

All about matter…this is what we were looking at in class Friday (A-day).  These and a number of other items can be found on the unit page for this unit: AP-1 –> Fall semester –> Unit 1

Remember that you will have a quiz on Wednesday (A) or Thursday (B) over material that we have covered since the start of school, specifically:

  1. Metric system and conversions between metric prefixes
  2. General dimensional analysis and conversions between units
  3. Applying orders of magnitude estimation
  4. The fundamental structure of matter

I planning the quiz to be about 20-25 minutes in length and I will call time.  Be ready, work diligently.

If you have not already done so, please set up your webassign account!  We will get some practice using webassign and learning about that system after the quiz.


AP-C: Calculus worksheet solutions

On the calculus exercises worksheet, please utilize the general forms of derivatives and integrals from the document in your packet (also found HERE)  On the derivatives worksheet you are to take the first and second derivatives of each function, however, at present, for our needs you may SKIP questions 5, 6, 7, and you should consider #10 to be your challenge problem.  Solutions for the first derivatives are below.  If you can take a first derivative, taking a second follows the same pattern, at least for simple polynomials.

For the integrals/anti-derivatives, you can skip items 4, 5, and 9.  Consider number 6 to be your challenge problem.  For info on how to work out the “definite” integrals on 7 and 8, see the end of the power point notes HERE  We will begin class on Wednesday with definite integrals just before we have our quiz.  Give these a try, but don’t freak out if you don’t get them.

Hopefully these will help you out.  If you are having difficulty with specific problems PLEASE email me or come in on Tuesday morning and let’s get it sorted out.  Also, I kind of rushed these so if you find something amiss please let me know ASAP.

Derivatives worksheet:

1) 21x^2      2) 6x+6     3) 30x^5-21x^2+3      4) -21x^6     8. re-write the function as x^-6, dy/dx = -6x^-7

9) let the function be written as 3x^(1/2),  dy/dx = (3/2)x^(-1/2)


Integration worksheet:

1) (4/2)x^2 + C = 2x^2 + C        2) x^3-3x^2+2x + C      3) x^4+(7/2)x^2 + C

7) anti-derivative = (4/3)x^3 + (3/2)x^2, now evaluate the anti-derivative at each limit and take the difference (top limit evaluation minus bottom), so:
(4/3)6^3 + (3/2)6^2 – 0 – 0
288 + 54 + 0 + 0

8. anti-derivative = ln x, now evaluate the anti-derivative at each limit and take the difference (top limit evaluation minus bottom), so:
ln5 – ln2 = ln5/ln2 = 2.32


Fare thee well, class of 2014!

Ladies and Gentlemen of the class of 2014:

It’s been super fun.  Really.  I’m not a big fan of goodbye’s, but I would be in denial if I told you that many of us would meet, together as a group, once more on this Earth.  You have been an amazing class, and through all of the ups and downs we have shared, I have learned some things from you even while, I certainly hope, you have learned some things from me.  I hope that besides physics, you have learned to question and to think.  I hope that you have come to enjoy irreverent sarcasm as much as I, and I hope that you have come, or at least are coming to enjoy the beauty of our world that lies beneath the numbers.  We live in an elegant place and I am mystified and fascinated by its nuances, captivated by its chaos, and captured by its order every day.  You, now, get to play a part in that (because apparently when you are 18 the social order decrees that you are ready, regardless of whether you were before or won’t be until later).  Enjoy your lives.  Keep in touch when you can.  Hit me up on Facebook…yes, that makes me old, but sometimes you need more than 140 characters to say something, and its much less of a pain than cataloging all of your email addresses which will surely change over the next 10 years.  I think I say this every year (except for the “cheating year,” and I don’t really talk about them very much), but for right now, you really have been the best, most successful class that I have ever taught, and I am very proud of you.

AP Seniors 2014


PreAP Spring Final Exam Information

PreAP Spring Final Exam Information



PreAP multiloop circuit practice “quiz” KEY

PreAP–KEY–multiloop circuits quiz


PreAP Combo circuit example from class (hard)

PreAP Combo circuit example from class

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