A heavy particle is projected from a point at the foot of a fixed plane, inclined at an angle $$450$$ to the horizontal, in the vertical plane containing the line of greatest slope through the point. If ∅ is the inclination to the horizontal of the initial direction of projection,For what value of $$tan$$∅ will the particle strike the plane horizontally andFor what value of $$tan$$∅ will the particle strike the plane at right angles?

Question

A heavy particle is projected from a point at the foot of a fixed plane, inclined at an angle $$450$$  to the horizontal, in the vertical plane containing the line of greatest slope through the point. If  ∅ is the inclination to the horizontal of the initial direction of projection,For what value of  $$tan$$∅ will the particle strike the plane horizontally andFor what value of $$tan$$∅ will the particle strike the plane at right angles?

Infinity Learn · Accepted Answer

Let the particle be projected from O with velocity u and strike the plane at a point P.   PQ $$=$$ OQ  as angle of incilantion is $$450$$ Maximum height    $$=$$ horizontal $$range2$$ value of    $$tan$$∅  so that the particle strike the plane horizontally $$u2sin2$$ϕ$$2g=u2sin2$$ϕ$$2g$$ $$u2sin2$$ϕ$$2g=u2sin$$ϕ$$cos$$ϕg      $$tan$$ϕ$$=2$$   At time   $$t=time$$ of $$flight=T=2usin($$ϕ−$$45o)gcos45o$$ component of velocity along the plane is zero $$v=u+at$$ , substitute the values, $$0=ucos($$ϕ−$$45o)$$−(gsin45o)t$$   $$ucos($$ϕ−$$45o)=(gsin45o)2usin($$ϕ−$$45o)(gcos45o) $$2tan($$ϕ−$$45o)=cot45o$$ $$2tan($$ϕ−$$45o)=1$$ ∴$$tan$$$$(A$$−$$B)=tanA$$−$$tanB1+tanAtanB$$    $$2tan$$ϕ−$$tan45o1+$$$$tan$$ϕ$$tan45o=1$$     ⇒  $$tan$$ϕ$$=3$$

Best Courses for You

Detailed Solution

courses

Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

download the app

Download the App