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On-a-frictionless-horizontal-surface-assumed-to-be-the-x-y-plane-a-small-trolley-A-is-moving-along-a-straight-line-parallel-to-the-y-axis-see-figure-with-a-constant-velocity-of-3-1-m-s-At-




Question Number 20425 by Tinkutara last updated on 27/Aug/17
On a frictionless horizontal surface,  assumed to be the x-y plane, a small  trolley A is moving along a straight  line parallel to the y-axis (see figure)  with a constant velocity of ((√3) − 1) m/s.  At a particular instant, when the line  OA makes an angle of 45° with the  x-axis, a ball is thrown along the  surface from the origin O. Its velocity  makes an angle φ with x-axis and it  hits the trolley.  1. The motion of the ball is observed from  the frame of the trolley. Calculate the  angle θ made by the velocity vector of  the ball with the x-axis in this frame.  2. Find the speed of the ball with  respect to the surface, if φ = ((4θ)/3)
Onafrictionlesshorizontalsurface,assumedtobethexyplane,asmalltrolleyAismovingalongastraightlineparalleltotheyaxis(seefigure)withaconstantvelocityof(31)m/s.Ataparticularinstant,whenthelineOAmakesanangleof45°withthexaxis,aballisthrownalongthesurfacefromtheoriginO.Itsvelocitymakesanangleϕwithxaxisandithitsthetrolley.1.Themotionoftheballisobservedfromtheframeofthetrolley.Calculatetheangleθmadebythevelocityvectoroftheballwiththexaxisinthisframe.2.Findthespeedoftheballwithrespecttothesurface,ifϕ=4θ3
Commented by Tinkutara last updated on 26/Aug/17
Answered by ajfour last updated on 28/Aug/17
(1.)  let the trolley move in the line  x=a and is hit by the ball in time t.  let speed of trolley be u,and that  of ball v.   atan φ=a+ut  asec φ=vt  ⇒  ((atan φ−a)/u)=((asec φ)/v)  or      vtan φ−v=usec φ        vsin φ−vcos φ=u  ⇒  ((vsin φ−u)/(vcos φ))=1 =tan θ   So θ=45°     (2.)    φ=((4θ)/3) =(4/3)(45°)=60°       ((vsin φ−u)/(vcos φ))=1 =tan θ   ⇒ ((v(√3))/2)−((√3)−1)=(v/2)  ⇒    v((√3)−1) = 2((√3)−1)           v= 2m/s .
(1.)letthetrolleymoveinthelinex=aandishitbytheballintimet.letspeedoftrolleybeu,andthatofballv.atanϕ=a+utasecϕ=vtatanϕau=asecϕvorvtanϕv=usecϕvsinϕvcosϕ=uvsinϕuvcosϕ=1=tanθSoθ=45°(2.)ϕ=4θ3=43(45°)=60°vsinϕuvcosϕ=1=tanθv32(31)=v2v(31)=2(31)v=2m/s.
Commented by Tinkutara last updated on 26/Aug/17
Thank you very much Sir!
ThankyouverymuchSir!ThankyouverymuchSir!
Commented by Tinkutara last updated on 28/Aug/17
Thank you!
Thankyou!Thankyou!

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