Mechanics, Calculus and 3D Vectors (David Morin\mathcal{David ~ Morin})

Problem Statement\textbf{Problem Statement} Attempt\mathbf{Attempt} Assume that the sphere is moving along the X axis.

Let the particle strike the sphere such that the radius vector of point of collision makes an angle θ\theta with the horizontal plane (the xy plane) and its projection on xy plane make an angle ϕ \phi with X axis. For the collision, the velocity of the sphere in the xy plane along the line of impact is vcosϕv \cos \phi . We have now reduced a 3D oblique collision into a 2D one. This velocity's projection on the line of impact is u=vcosϕcosθ\boxed{u=v \cos \phi \cos \theta} . Thus the collision takes place along the radius vector whose direction is r^=cosϕcosθi^sinϕcosθj^+sinθk^ \hat{r}=\cos \phi \cos \theta \hat{i} - \sin \phi \cos \theta \hat{j} + \sin \theta \hat{k} ow since the sphere is large, final velocity of particle along r^\hat{r} is 2u2u, and along the X direction, the total change in momentum of particle is ΔPx=mux=2mvcos2ϕcos2θ\Delta P_x= mu_x=\boxed{2mv \cos^2 \phi \cos^2 \theta} Now the volume swept out by the region in time Δt\Delta t is dV=vR2Δtcosϕcosθ dϕ dθdV= vR^2 \Delta t \cos \phi \cos \theta { ~d \phi }{ ~d \theta } and the number of particles hitting is ndVn \cdot dV, thus the total force acting on the sphere along X axis is F=2mv2nR2π/2π/2π/2π/2cos3ϕcos3θ dϕ dθI.F= 2mv^2nR^2 \color{#3D99F6}{\underbrace{\color{#D61F06}{\displaystyle \int_{-\pi /2}^{\pi /2} \int_{-\pi /2}^{\pi /2} \cos^3 \phi \cos^3 \theta ~d \phi ~d \theta}}_{I}}. We can integrate over ϕ\phi for a particular θ\theta and then integrate over θ\theta. So F=329mv2nR2\boxed{F=\dfrac{32}{9}mv^2nR^2}. Official Solution\textbf{Official Solution} I don't quite get how the theta (θ)(\theta) is taken here, (from which axis or plane) and also the answer obtained is different from mine by a numerical factor ( 32/9 Vs π\pi). Please help on this.

Mechanics #Momentum #JEEPhysics #Calculus #Physics #DoubleIntegrals

Note by Harsh Poonia
10 months, 3 weeks ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

@Gordon Chan @Steven Chase @Mark Hennings sir please help.

Harsh Poonia - 10 months, 3 weeks ago

Log in to reply

Lol a callout for help for the best physicists and engineers on (not to mention Karan Chatrath)

Krishna Karthik - 3 months, 2 weeks ago

Log in to reply

@Md Zuhair please help.

Harsh Poonia - 10 months, 2 weeks ago

Log in to reply

Sorry bro... I have not checked Brilliant from 4 months now! Sorry to see this late...

Md Zuhair - 8 months, 1 week ago

Log in to reply

Its ok. Any inputs on the problem?

Harsh Poonia - 8 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...