Overcoming Difficult 2D Motion Problems: My Journey to Advanced Physics

[bigmike94]

Ive been reading University physics by roger freedman, I’m on section 3 motion in 2d. I can solve most of the problems ag the end of the chapter, or at least understand the solutions.

But there is a small extra section called challenge problems. There’s only 3 but I found them very difficult, so much so I couldn’t actually solve them. I've posted the question, even using the instructors solution manual I didn’t understand it 100%, I could follow it a little.

Bare in mind I’ve not just read this book regarding 2d motion. I've watched a few University lectures and plenty of other videos, I felt really comfortable with 2d motion and with the normal problem set.

Should I be disheartened that I couldn’t solve this type of question? I never come across this type of problem in all the hours I’ve put into 2d motion.

My goal is to become proficient in topics like particle physics, thermodynamics, electromagnetism and quantum. I study extremely hard.

Did anyone else have issues with problems like this but still went on to do really well in more advanced topics? Should I ask a private tutor to give me similar problems and walk me through the steps taken?

 

[PeroK]

What amounts to a challenge depends on how advanced your abilities are. That problem looks a bit complicated to understand, but ultimately is elementary, I would say. Slightly tricky, perhaps.

PS One issue might be that you have to solve it algebraically. It would be hard if you are used to simply plugging in numbers.

 

[BvU]

Hello best dad @bigmike94 ,

$\qquad$ !​

Should I be disheartened that I couldn’t solve this type of question?

I wouldn't worry too much if I were you. To me it looks like an exercise where the main task is to unearth the actual physics problem that's wrapped up in a long story. A little sorting should carry a long way, as would a clear drawing of the situation evolving.

I find

I can solve most of the problems at the end of the chapter, or at least understand the solutions.

both encouraging and worrying. Don't look in the solutions manual too quickly !

It looks as if you are self-studying. If so: respect ! And doing the exercises is the best you can do.

By the way, my son's Freedman is 11th ed. and it has 8 challenges 3.86-3.93. Yours is 3.92.

$\$

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[jbriggs444]

Should I be disheartened that I couldn’t solve this type of question? I never come across this type of problem in all the hours I’ve put into 2d motion.

It it not a difficult problem. You might benefit from attacking it and gaining assistance in the introductory physics homework sub-forum.

It is an example of a problem where you are handed too much information and need to ignore the irrelevant. You cannot just do a pattern match for the canned formula that features the knowns you have and the unknown that you want.

As to feeling disheartened... Solving physics problems is easy. Making life choices is hard.

 

[bigmike94]

What amounts to a challenge depends on how advanced your abilities are. That problem looks a bit complicated to understand, but ultimately is elementary, I would say. Slightly tricky, perhaps.

PS One issue might be that you have to solve it algebraically. It would be hard if you are used to simply plugging in numbers.

Just plugging in numbers isn't me. I like to fully understand the situation. I sat for nearly two hours trying to understand this problem mathematically. I have posted the solution below. It’s not as easy as most folks might think it is

 

[bigmike94]

Hello best dad @bigmike94 ,

$\qquad$ !​

I wouldn't worry too much if I were you. To me it looks like an exercise where the main task is to unearth the actual physics problem that's wrapped up in a long story. A little sorting should carry a long way, as would a clear drawing of the situation evolving.

I find

both encouraging and worrying. Don't look in the solutions manual too quickly !

It looks as if you are self-studying. If so: respect ! And doing the exercises is the best you can do.

By the way, my son's Freedman is 11th ed. and it has 8 challenges 3.86-3.93. Yours is 3.92.

$\$

$\$

Thank you for your reply. Yes I’m self studying. My physics level is beginner because I wanted to get to a calculus level maths before I started. Which I had to learn pre algebra and algebra myself first. So it’s been quite a journey. I definitely won’t give up. It just threw me off. Once again I appreciate your reply

 

[PeroK]

That "book" solution is a bit plug-and-chuggish to me.

I must admit, the first thing I would do is change to the reference frame of the airliner. Then, the airliner is at rest and the rocket is at rest initially. Then:

1) The rocket falls for some time $T$. At the end of which time it is some distance below the rocket and falling vertically at some speed.

2) The rocket then accelerates in two dimensions. So, we write down the equations of motion in the $x$ and $y$ directions. I would start again with $t = 0$ at this point.

3) The rocket must travel a horizontal distance of $1 \km$ before reaching the level of the aircraft ($y = 0$, say). That gives us $t$ (or an expression for $t$).

4) We need $y = 0$ at this time $t$. That gives us a quadratic to solve.

Perhaps trickier than I first thought!

 

[bigmike94]

That "book" solution is a bit plug-and-chuggish to me.

I must admit, the first thing I would do is change to the reference frame of the airliner. Then, the airliner is at rest and the rocket is at rest initially. Then:

1) The rocket falls for some time $T$. At the end of which time it is some distance below the rocket and falling vertically at some speed.

2) The rocket then accelerates in two dimensions. So, we write down the equations of motion in the $x$ and $y$ directions. I would start again with $t = 0$ at this point.

3) The rocket must travel a horizontal distance of $1 \km$ before reaching the level of the aircraft ($y = 0$, say). That gives us $t$ (or an expression for $t$).

4) We need $y = 0$ at this time $t$. That gives us a quadratic to solve.

Perhaps trickier than I first thought!

Thats a really good way to look at it! And yes that’s what I thought. When I first read it I thought I’d have it solved in no time. 2 hours later i was nearly pulling my hair out I am going to do the following chapters Which are all about Newton’s laws etc then maybe come back to the problem and tackle it again

 

[PeroK]

Thats a really good way to look at it! And yes that’s what I thought. When I first read it I thought I’d have it solved in no time. 2 hours later i was nearly pulling my hair out I am going to do the following chapters Which are all about Newton’s laws etc then maybe come back to the problem and tackle it again

This is less a problem that tests your knowledge of theory than your ability to manage a longer problem with many more steps to conclusion. You need to develop the mental foresight and stamina.

It's a bit like you start by looking at chess puzzles that have a checkmate in 1-2 moves. Then you move on to puzzles that are checkmate in 3-4 moves.

Or, when you start writing longer computer programs: it's not the syntax but the ability to organise a lot more steps and data.

 

[bigmike94]

This is less a problem that tests your knowledge of theory than your ability to manage a longer problem with many more steps to conclusion. You need to develop the mental foresight and stamina.

It's a bit like you start by looking at chess puzzles that have a checkmate in 1-2 moves. Then you move on to puzzles that are checkmate in 3-4 moves.

Or, when you start writing longer computer programs: it's not the syntax but the ability to organise a lot more steps and data.

I suppose that will come with time. I have definitely painted the wrong picture of myself here. I always spend a great deal of time on every question to fully understand what’s going on. I don’t like plugging in numbers and I actually try derive the equations I need if I can. For example I was doing derivatives of inverse trig function a couple of days ago and with each problem I derived the formula instead of plugging in the numbers. It takes away the fun in my opinion. I’m not sure what this says about me regarding physics but I see it as a good sign

 

[PeroK]

I suppose that will come with time. I have definitely painted the wrong picture of myself here. I always spend a great deal of time on every question to fully understand what’s going on. I don’t like plugging in numbers and I actually try derive the equations I need if I can. For example I was doing derivatives of inverse trig function a couple of days ago and with each problem I derived the formula instead of plugging in the numbers. It takes away the fun in my opinion. I’m not sure what this says about me regarding physics but I see it as a good sign

Let me have a proper look at this problem.

 

[PeroK]

Here's my "algebraic" solution. We are using the frame of the plane. Only the rocket moves. First, it falls for time $T$ and this will give us our initial $y$ coordinate and initial component of velocity in the $y$ direction:$$y_0 = -\frac 1 2 gT^2, \ u_y = -gT$$During the second phase, where we wil start again at $t = 0$, the rocket must travel $d = 1 km$ horizontally and get back to $y = 0$ vertically The rocket's acceleration is:
$$a_x = 3g\cos(30) = \frac{3\sqrt 3 g}{2}, \ a_y = 3g\sin(30) = \frac{3g}{2}$$We can solve for the motion in the $x$ direction:
$$d = \frac 1 2 a_x t^2 \ \Rightarrow \ t = \sqrt{\frac{4d}{3\sqrt 3 g}} \approx 8.86s$$And, we can solve for motion in the $y$ direction: $$0 = y_0 + u_yt + \frac 1 2 a_y t^2 \ \Rightarrow \ -\frac 1 2 gT^2 -gTt + \frac 1 2(\frac{3g}{2})t^2 = 0 $$Note that $g$ cancels and we can tidy up that quadratic:
$$T^2 +2tT - \frac 3 2 t^2 = 0$$ which has the positive solution:$$T = (\sqrt{\frac 5 2} - 1)t \approx 5.15s$$And, in fact, we have a general formula, where $d$ is the distance (clearance) in front of the plane:
$$T = (\sqrt{\frac 5 2} - 1)\sqrt{\frac{4d}{3\sqrt 3 g}}$$ If you want to be clever, you could generate a formula for an general angle $\theta$!

 

[PeroK]

PS that's always the way I've done things. I've never done things the way it's done in the book! Why not use the single letter $g$ instead of $9.8 ms^{-2}$?

Also, I think my solution is a lot easier to check for errors.

 

[bigmike94]

$$a_x = 3g\cos(30) = \frac{3\sqrt 3 g}{2}, \ a_y = 3g\sin(30) = \frac{3g}{2}$$We can solve for the motion in the $x$ direction:
$$d = \frac 1 2 a_x t^2 \ \Rightarrow \ t = \frac{4d}{3\sqrt 3 g} \approx 8.86s$$And, we can solve for motion in the $y$ direction: $$0 = y_0 + u_yt + \frac 1 2 a_y t^2 \ \Rightarrow \ -\frac 1 2 gT^2 -gTt + \frac 1 2(\frac{3g}{2})t^2 = 0 $$Note that $g$ cancels and we can tidy up that quadratic:
$$T^2 +2tT - \frac 3 2 t^2 = 0$$ which has the positive solution:$$T = (\sqrt{\frac 5 2} - 1)t \approx 5.15s$$And, in fact, we have a general formula, where $d$ is the distance (clearance) in front of the plane:
$$T = (\sqrt{\frac 5 2} - 1)\frac{4d}{3\sqrt 3 g}$$ If you want to be clever, you could generate a formula for an general angle $\theta$!

I have read through your reply 4/5 times. It’s starting to sink in, I’m going to go through it again later with whiteboard and pen to fully try understand. The books solution was very difficult to read for me so I appreciate you taking the time to solve it. I don’t think I have the skill set to have solved this one by myself, I knew there would be two different times involved somehow but couldn’t figure it out. I really thought I was ready to move onto the next chapter as like I said I can solve the rest of the problems but not the challenge problems.

I don’t think I was the first and won’t be the last student to find a problem hard but after spending a good 5 month on kinematics I think it’s time I moved onto Newton’s laws

Thanks once again for your help

 

[bigmike94]

PS that's always the way I've done things. I've never done things the way it's done in the book! Why not use the single letter $g$ instead of $9.8 ms^{-2}$?

Also, I think my solution is a lot easier to check for errors.

Last question there is a similar calculus based intro physics book which is similar length to freedman one. I was thinking once I finally finish this book. Would it be worth going through the other one to solidify my learning. Or does it not work like that? Is it just a linear process where we keep using and building upon past topics that it would be redundant to start again from the beginning?

 

[PeroK]

Last question there is a similar calculus based intro physics book which is similar length to freedman one. I was thinking once I finally finish this book. Would it be worth going through the other one to solidify my learning. Or does it not work like that? Is it just a linear process where we keep using and building upon past topics that it would be redundant to start again from the beginning?

I'm not sure what's best, to be honest.

 

[Mondayman]

Last question there is a similar calculus based intro physics book which is similar length to freedman one. I was thinking once I finally finish this book. Would it be worth going through the other one to solidify my learning. Or does it not work like that? Is it just a linear process where we keep using and building upon past topics that it would be redundant to start again from the beginning?

Physics for Scientists and Engineers by Serway is almost identical. I think it's tad better written, and the problems tend to have more diagrams. I have both, and use both right now to reintroduce myself to solving physics problems, as I have been out of school for a few years.

I would not recommend working through another entire intro physics book when you finish Freedman. I would move onto something more challenging, but only when you are confident in your ability to solve the problems in the Freedman/Serway text. I only have both because I needed one for school, and wanted another massive resource of problems to work through.

 

[bigmike94]

Physics for Scientists and Engineers by Serway is almost identical. I think it's tad better written, and the problems tend to have more diagrams. I have both, and use both right now to reintroduce myself to solving physics problems, as I have been out of school for a few years.

I would not recommend working through another entire intro physics book when you finish Freedman. I would move onto something more challenging, but only when you are confident in your ability to solve the problems in the Freedman/Serway text. I only have both because I needed one for school, and wanted another massive resource of problems to work through.

Thank you for your reply! i am looking forward to the more advanced topics. But I want the basics mastered first. I don’t want to be left high and dry. Similar happened with maths. I was skipping too any topics and it came back to bite me. I find myself redoing calc 2 and enjoying it. I recommend to anyone profesor Leonard lectures on YouTube

 

[PeroK]

If the acceleration of the rocket is $kg$ at an angle $\theta$ above the horizontal, then:

Spoiler

$$T = (\sqrt{1 + k\sin \theta} - 1)\sqrt{\frac{2d}{kg\cos \theta}}$$

 

[Haborix]

I want to add a general comment about the degree of skill and ability that should be acquired before moving to the next topic. One could never get past past classical mechanics if perfect mastery was required---there are always more complicated and convoluted problems. Second, what often "solves" a tricky problem is the time after you initially try to solve it. There are no doubt problems I could solve quickly now that I would not have been able to solve in a reasonable amount of time when I first learned the subject. So there is a balancing act. When you sit in a structured course that balancing act is largely done for you, but when you self-study you have to do it on your own. In my opinion, that is what makes self-study incredibly challenging.

I hope this isn't off topic, but it seemed to be an implicit issue within the original post.

 

[bigmike94]

I want to add a general comment about the degree of skill and ability that should be acquired before moving to the next topic. One could never get past past classical mechanics if perfect mastery was required---there are always more complicated and convoluted problems. Second, what often "solves" a tricky problem is the time after you initially try to solve it. There are no doubt problems I could solve quickly now that I would not have been able to solve in a reasonable amount of time when I first learned the subject. So there is a balancing act. When you sit in a structured course that balancing act is largely done for you, but when you self-study you have to do it on your own. In my opinion, that is what makes self-study incredibly challenging.

I hope this isn't off topic, but it seemed to be an implicit issue within the original post.

Thank you for your input I really do appreciate the replies, as I’ve said previously I feel comfortable with over 95% of the problems. So I shouldn’t beat myself up about it. I’m quite bad for putting myself down.

On the plus side you was also correct about giving it time, believe it or not I struggled to sleep last night because the problem was just circling in my head I was thinking of ways I could solve it without directly copying solutions. I needed my own way.

So I got home after work with my whiteboard and pen and really thought about it. I wrote down all the knows and unknowns and motion equations changed the reference frame and created two times etc and FINALLY I managed to solve it!

What a feeling. Now I am moving onto the next chapter in the textbook feeling satisfied