Trajectory Properties

Classical Mechanics Level 3

A projectile is launched from ground level with speed $v_0$ at an angle $\theta$ with respect to the horizontal. The ambient gravitational acceleration is $g$ . Suppose we make a plot of the horizontal position $(x)$ vs. the vertical position $(y)$ . The plot includes all points for which $y \geq 0$ .

Let the length of the resulting curve be $L$ , and let the area between the curve and the $+x$ axis be $A$ . What is the product of $L$ and $A$ ?

Details and Assumptions:
1) $v_0 = 20$
2) $\theta = \pi/4$
3) $g = 10$
4) This is a good problem for practicing numerical techniques

The answer is 12243.1.

2 solutions

Nj Star
Mar 11, 2020

Yup, I did it the exact same way.

Krishna Karthik - 1 year, 2 months ago

Hey are you Neeraj Anand Badgujar? If so, bro, why did you delete your account? Just curious. Steven Chase tells me "some drama happened" earlier. I wonder.

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik i was just giving a advice to him to, not to post solutions of easy problem and there are plenty of peoples in brilliant to post that little solutions.
Because I always consider him as a very intelligent person.
Instead of accepting compliment, he very little bit scolded me(I think he felt a little bad)

After that I was feeling nervous and bad. Therefore in a hurry I deleted my account.
I am also willing to say sorry, but I don't think he will forgive me .

NJ STAR - 10 months, 4 weeks ago

Why did you delete your account? So many contributions that you made... why did you take what he said so seriously?

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik because I always consider him as a series guy.

NJ STAR - 10 months, 4 weeks ago

@Nj Star – But what exactly did you both say? From what I asked him, he said "I won't go into the details"; this implies that he must've got pretty annoyed.

Krishna Karthik - 10 months, 4 weeks ago

@Nj Star – And I don't think he would've gotten riled up for such a small comment. Either-way, don't take it too personally. This is just the internet mate.

Krishna Karthik - 10 months, 4 weeks ago

What did he say to you when he got annoyed bro?

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik He just said “if you keep behaving this way, I am going to completely ignore you”

NJ STAR - 10 months, 4 weeks ago

@Nj Star – Bruh that's cold as fuck

Krishna Karthik - 10 months, 4 weeks ago

@Nj Star – That is so cold... he must've got really pissed. lmao

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik what does cold mean?

NJ STAR - 10 months, 4 weeks ago

@Nj Star – "rude", "blunt", "lack of warmth" could work as similar words to "cold". What did you say to him that made him get this way? I mean... did your comment about his solutions alone make him say this?

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik other than this I only commented one thing,in that time I was not serious, that is
“Now you have read the comment but you don't want to reply”
The main comment is the above comment which I have already said.

NJ STAR - 10 months, 4 weeks ago

@Nj Star – Yeah. I've figured it out. Sometimes you get a little pushy when you want help with people like Steven Chase or Karan Chatrath or some others. It's ok, but sometimes people look at it like it's offensive.

He probably saw it as offensive because you seemed a bit pushy. Give people time to help you.

Please don't leave brilliant bro.

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik Hmm, I agree with you.

NJ STAR - 10 months, 4 weeks ago

@Nj Star – Is your main account permanently deleted? Is there a way which you could reclaim it? It's a pity it's gone...

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik I have done that 2nd option, even staff can't do anything.

NJ STAR - 10 months, 4 weeks ago

@Nj Star – It's ok bro, you can keep posting problems and doing stuff with this account. I hope you have a good time on brilliant.

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik I will only do that if Steven sir will forgive me for my mistake.

NJ STAR - 10 months, 4 weeks ago

@Nj Star – yeah yeah; he'll forgive you. Just say that you won't be pushy anymore and say that you will understand if he can't help/won't help. Alright. Good luck my man.

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik and I don't think in my previous Id there are many contributions.
I have enough enormous energy that I can post thousands of original problem with my own in 1 day.
I just deleted because I was feeling nervous at that time.

NJ STAR - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik BTW how do you know that I am Neeraj??

NJ STAR - 10 months, 4 weeks ago

@Nj Star – Because "NJ" comes before your Chess.com account too. Your chess.com account name is "NJ Physicist". So you must be Neeraj. Also, your handwriting (see the above solution) is similar to some of the handwritten solutions by "Neeraj".

Krishna Karthik - 10 months, 4 weeks ago

@Krishna Karthik – @Krishna Karthik But this solution is 4 months old.
Don't mind,I think someone has told you.?

NJ STAR - 10 months, 4 weeks ago

@Nj Star – No. Just simple deduction😄 I am being very truthful. "NJ" comes before your chess.com name. And you commented on the EM discussion with Steven Chase. So many correlations between you and Neeraj. So I simply deduced that you were Neeraj.

Krishna Karthik - 10 months, 4 weeks ago

@Nj Star – And you do tend to give solutions to a lot of Steven Chase's problems. btw both accounts' handwriting is too similar to be a coincidence.

Krishna Karthik - 10 months, 4 weeks ago

Bro... you shouldn't have deleted your account. And please tell me exactly what you said to him. Btw if he gets annoyed, it's not your problem. You should just tell him that you meant no offence.

Krishna Karthik - 10 months, 4 weeks ago

Krishna Karthik
Apr 10, 2020

Here is a basic simulation with linear extrapolation:

import math

#Numerical integration with solutions attached below.

#initialisation 
time = 0
curveLength = 0
deltaT = 0.00001
Area = 0
distanceX = 0
distanceY = 0

#numerical integration of velocity by dt; curve length and area
while  time <= 2*math.sqrt(2): 
  Dx = (20/math.sqrt(2))*deltaT # dx

  distanceY = (20/math.sqrt(2))*time - 5*time**2 #y
  Area = Area + distanceY*Dx # A = y dx

  velocityMagnitude = 10*(math.sqrt(time**2 - 2*math.sqrt(2)*time +4)) #magnitude of velocity
  Ds = velocityMagnitude*deltaT #distance travelled in small time     
  curveLength = curveLength + Ds #updating curvelength value by adding Ds
  time = time+deltaT #updating time value for next iteration

print("Length of trajectory path: " +str(curveLength))
print("Area under trajectory path: "+str(Area))

#Length of trajectory path: 45.911800
#Area under trajectory path: 266.66666

Trajectory Properties

The answer is 12243.1.

2 solutions

0 pending reports