Using vectors, show that r(t) lies in a plane where r(t) shows projectile motion
ID: 3118784 • Letter: U
Question
Using vectors, show that r(t) lies in a plane where r(t) shows projectile motion.
r rightarrow (t) = v rightarrow 0 + v rightarrow 0 t + 1/2 g rightarrow t2Explanation / Answer
lets consider the plane containing vo and g the normal to that plane will be given by vo x g Now, if r(t) is always in this plane , then it must also be normal to this vector => r(t) . (vo x g) must be zero Checking => r(t) . (vo x g) = (r(t) x vo) . g [ using triple product property a .(b x c) = (a x b). c] = ( {vo + vo*t + (1/2)g*t^2} x vo) . g = {(vo x vo) + (vo x vo)*t + (1/2)(g x vo)*t^2} . g = ((1/2)(g x vo)*t^2) . g [ using a x a = 0 ] = ((t^2)/2){ (g x vo) . g } = ((t^2)/2){ (g x g) . vo } [ using triple product property (a x b). c = (c x a). b] = 0 [ using a x a = 0 ] Thus, r(t) lies in the plane where r(t) shows projectile motion
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