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
"v(t)=r'(t)"
"=2t\\sinh (t^2-1)i- \\cosh(1-t)j+2Btk"
"=(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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