How to approximate—kinematic question using special cases

[Demetrius]

Homework Statement

A ball is thrown at an angle Θ up to the top of a cliff of height L, from a point a distance L, from the base, as shown in the figure below. Assuming that one of the following quantities is the initial speed required to make the ball hit right at the edge of the cliff, which one is it? (Do not solve problem from scratch, Just check special cases)

Homework Equations

Here are the following quantities the answer may be:

$$

\sqrt {\frac {gL} {2(tan(\theta)-1)}} , \frac 1 {cos(\theta)} \sqrt {\frac {gL} {2(tan(\theta) - 1)}} , \frac 1 {cos(\theta)} \sqrt {\frac {gL} {2(tan(\theta) + 1)}} , \sqrt {\frac {gLtan(\theta)} {2(tan(\theta) + 1)}}

$$

The Attempt at a Solution

I need to determine out of the given expressions, which one is correct for the initial speed. All the given quantities have the correct units and they all behave correctly as g grows and as L grows. The remaining input quantity that needs to be checked is Θ. There are three cases where Θ need to be checked.

Case 1: As Θ → 0 , v → $C\sqrt {gL}$ where C is a dimensionless number

Case 2: As Θ → $pi/4$ , v → $C\sqrt {gL}$ where C is a dimensionless number

Case 3: As Θ → $pi/2$ , v → ∞

For case 1, I used dimensionless analysis to determine the expression for v

For case 3, I reasoned that as Θ becomes steeper than the the velocity will need to increase to compensate

For case 2, I made a guess that v should be an expression similar to case 1 however I can not prove/show it. I just believe it needs to be a number

Quantity one does not pass case 3. Quantity 4 does not pass case 1.Quantity 2 does not pass case 2.
So, I chose quantity 3 as my answer.

Is my reasoning for case 2 correct? I cannot prove it. I really just guess. I want to know what will happen as Θ → $pi/4$? And did I chose the wrong quantity?

 

[haruspex]

Quantity 4 does not pass case 1.Quantity 2 does not pass case 2.

I do not understand how you can use dimensional analysis to rule out any options. All the options are dimensionally correct.
You mentioned θ→π/4. That is indeed an important test. Try to decide.

 

[kuruman]

What if you derived the equation of the parabolic path to get y as a function of x then set x = y = L and solve for the speed? Then you will verify whether it is case 2 or not.

 

[haruspex]

What if you derived the equation of the parabolic path to get y as a function of x then set x = y = L and solve for the speed? Then you will verify whether it is case 2 or not.

That would be contrary to the instructions.

 

[kuruman]

That would be contrary to the instructions.

Indeed, but if one knows the correct answer and why it is correct, then one's thinking can be informed in deciding why the other answers are incorrect. I don't think that's cheating if the final answer is presented as instructed.

 

[haruspex]

Indeed, but if one knows the correct answer and why it is correct, then one's thinking can be informed in deciding why the other answers are incorrect. I don't think that's cheating if the final answer is presented as instructed.

Arguably it is cheatng, but perhaps more importantly, in an exam, not enough time would be allowed for such an approach. The point of the exercise is to practise ways of sanity-checking answers.

 

[Demetrius]

Well, did I chose the right one? Is quantity 3 the correct expression? My reasoning being that it behaves as expected in all three cases

 

[haruspex]

Well, did I chose the right one? Is quantity 3 the correct expression? My reasoning being that it behaves as expected in all three cases

Option 2 also passes all your tests. Only one of them gives the right asymptotic C as θ→π/4.

 

[Demetrius]

Yes, option 2 does pass all my checks. But as option 2 approach $pi/4$ the denominator will eventually become zero resulting in the expression going to infinity. And v should not go to infinity as $theta$ go to $pi/4$.

For the above reason, this is why I chose option 3.

I do not know if my reasoning above is logical or algebraically correct but this was my thought process

 

[haruspex]

Yes, option 2 does pass all my checks. But as option 2 approach pi/4 the denominator will eventually become zero resulting in the expression going to infinity. And v should not go to infinity as θ→π/4.

Think about that some more.

 

[Demetrius]

Wait, I do not understand. As Θ→ $pi/4$ , v should be some expression in the form $C \sqrt {gL}$ where C is a dimensionless number. This is what I believe. Are you saying this is wrong? Or are you saying my reasoning in my previous post was incorrect?

And again thanks for the help, I appreciate your patience. I don't want to come off as rude, I just want to learn

 

[haruspex]

Wait, I do not understand. As Θ→ $pi/4$ , v should be some expression in the form $C \sqrt {gL}$ where C is a dimensionless number.

Sure, but as I wrote, all four options pass that test if you allow infinity as a value for C. The question is, what should the asymptotic value of C be as Θ→ $pi/4$.

 

[Demetrius]

Okay, now I am starting to understand. Thanks. My goal is to find a value of C. I did not know I could find a value for C with the given information. So How should I go about this? What steps should I take? Is there some particular method or do I reason through it?

 

[haruspex]

My goal is to find a value of C.

Not specific values in general. It's just a question of whether it is finite. Some of the options give finite values, some don't. Should the value be be finite or infinite? Which options fit that?

 

[Demetrius]

Okay, I believe that C should be finite when 0 ≤ Θ < $pi/2$. Once Θ = $pi/2$ C should be infinite.

 

[haruspex]

Okay, I believe that C should be finite when 0 ≤ Θ < $pi/2$. Once Θ = $pi/2$ C should be infinite.

How will it reach the target if θ<π/4?

 

[Demetrius]

I am starting to understand. Ok, it will not reach the target if Θ < $pi/4$. Then will the value of C be infinite for Θ < $pi/4$?

 

[haruspex]

I am starting to understand. Ok, it will not reach the target if Θ < $pi/4$. Then will the value of C be infinite for Θ < $pi/4$?

For θ<π/4, even a straight trajectory will not get there. Likely the equations would produce an imaginary result. What about θ=π/4?

 

[Demetrius]

I did not consider that the result could be imaginary. Nevertheless, to answer your question:

Θ = $pi/4$ then C should be a finite number.

 

[haruspex]

I did not consider that the result could be imaginary. Nevertheless, to answer your question:

Θ = $pi/4$ then C should be a finite number.

Right! So which option matches that?

Edit: I misread your post. I thought you wrote "infinite"

 

[Demetrius]

I still conclude that option 3:

$$
\frac 1 {cos(\theta)} \sqrt {\frac {gL} {2(tan(\theta) + 1)}}
$$
is the correct answer. When I plug in $\theta=pi/4$ the result was a finite number.

However when I plug in $\theta=pi/4$ for option 2 the result was infinite.

Am I wrong? If so, What am I doing wrong? Should I not be plugging in?

 

[haruspex]

I still conclude that option 3:

$$
\frac 1 {cos(\theta)} \sqrt {\frac {gL} {2(tan(\theta) + 1)}}
$$
is the correct answer. When I plug in $\theta=pi/4$ the result was a finite number.

However when I plug in $\theta=pi/4$ for option 2 the result was infinite.

Am I wrong? If so, What am I doing wrong? Should I not be plugging in?

Sorry, I misread your post #19. Please see my edited post #20.

 

[Demetrius]

No worries, I am learning thanks to you. And I appreciate your patience.

So, since C should be infinite when $\theta = pi/4$. The answer should be option 2.

I just need to understand why C should be infinite? And I honestly I have no Idea why C is infinite when $\theta = pi/4$?

Can you please offer an explanation?

 

[haruspex]

No worries, I am learning thanks to you. And I appreciate your patience.

So, since C should be infinite when $\theta = pi/4$. The answer should be option 2.

I just need to understand why C should be infinite? And I honestly I have no Idea why C is infinite when $\theta = pi/4$?

Can you please offer an explanation?

In order to hit the target at pi/4, what must the trajectory look like?

 

[Demetrius]

At $\pi/4$ the trajectory must be a straight line. Right? If it is at $\pi/4$ then the projectile is pointing directly at the edge of the cliff. Therefore it must take a straight path.

This is correct, right?

 

[haruspex]

At $\pi/4$ the trajectory must be a straight line. Right? If it is at $\pi/4$ then the projectile is pointing directly at the edge of the cliff. Therefore it must take a straight path.

This is correct, right?

Right. What speed must it go at to travel in a straight line, despite gravity?

 

[Demetrius]

To travel in a straight line then the speed must be infinite.

If the speed was not infinite then the trajectory will be parabolic because of gravity.

Is this correct? This makes sense to me but I been wrong for most of the day. So, I'm not as confident

 

[haruspex]

To travel in a straight line then the speed must be infinite.

If the speed was not infinite then the trajectory will be parabolic because of gravity.

Is this correct? This makes sense to me but I been wrong for most of the day. So, I'm not as confident

You have it right now.

Notice what happens if you put θ<π/4 in option 2. This result is not unusual for circumstances that are beyond even the limiting case of physical possibility.

 

[Demetrius]

Wow, that was a journey. My brain is still processing everything but I appreciate your patience and help. Thanks for everything

Also, do you have any general advice or tips that will be helpful for me when I face similar problems in the future?

 

[haruspex]

do you have any general advice or tips that will be helpful for me when I face similar problems

You started off looking to use dimensional analysis. That is often a good approach with such problems, but you should have checked first whether any of the offered answers were dimensionally different. In this case, they weren't, so dimensional analysis was never going to be a way of ruling some out.
Next, you had the right idea, to consider special cases, and you did pick out the two of interest, π/2 and π/4. Unfortunately you missed that for π/4 the answer should be infinite.