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
3 weeks ago

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

Harsh Poonia - 3 weeks ago

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@Md Zuhair please help.

Harsh Poonia - 2 weeks, 5 days ago

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