Answer to Question #233707 in Differential Geometry | Topology for JOHNNIE

Question #233707

Let the position vector of a stone at time,t be given as r(t)=cosh (t^2-1)I+ sinh(1-t)j+Bt^2k

Assume that the position vector is Normal to the acceleration vector.

Find the value of B at Time t=5seconds


1
Expert's answer
2021-09-07T02:34:44-0400
"r(t)=\\cosh (t^2-1)i+ \\sinh(1-t)j+Bt^2k"

"v(t)=r'(t)"

"=2t\\sinh (t^2-1)i- \\cosh(1-t)j+2Btk"


"a(t)=r''(t)"

"=(2\\sinh (t^2-1)+4t^2\\cosh (t^2-1))i+ \\sinh(1-t)j+2Bk"

"r(t)\\cdot a(t)=0"

"\\cosh (t^2-1)(2\\sinh (t^2-1)+4t^2\\cosh (t^2-1))"

"+ \\sinh(1-t) \\sinh(1-t)+Bt^2(2B)=0"

"B^2t^2=-\\cosh (t^2-1)\\sinh (t^2-1)"

"-2t^2\\cosh^2 (t^2-1)-\\sinh^2(1-t)"

"t=5"

"25B^2=-\\cosh (25-1)\\sinh (25-1)"

"-2(25)\\cosh^2 (25-1)-\\sinh^2(1-5)"

"B^2=\\dfrac{1}{25}(-\\dfrac{\\sinh(48)}{2}-50\\cosh^2 (24)+\\sinh^2(4))<0"

There are no solution for "B\\in \\R."


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