For the one-dimensional motion, described by x = t - sint

1.	For the one-dimensional motion, described by x = t - sint
Answer» `X (t) gt 0` for all `t gt 0` `V (t) gt 0` for all` t gt 0` `a (t) gt 0` for all `t gt 0` v (t) LIES between 0 and 2 Solution :Given, `x =t - sin t ` velocity `v = (dx)/(dt) = (d)/(dt) [t - sin t]` `=1 - COS t` Acceleration `a = (DV)/(dt) = (d)/(dt) [1- cos t] sin t ` As acceleration `a gt 0 ` for all `t gt 0 (because sin t gt 0)` Hence,` x (t) gt 0` for all `t gt 0` Velocity `v =1 - cos t` When, `cos t = 1,` velocity `v =0` `V_(max) =1 - (cos t ) _(min) =1 - (-1) =2` `V _(min) =1- (cos t) _(max) 1-1 =0` Hence, v lies between 0 and 2. Acceleration `a - (dv)/(dt) =- sin t` When `t =0, x=0, v =+ 1, a =0` When `t = 90^(@),x = 1, v =0, a =-1` When `t = 180^(@) , x=0, v =-1, a =1` When `t = 360 ^(@),x =0 , v =0 ,a=0`

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