i have calc 3 online hw its 4 questions and i need help with it

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1)Find the length of the curve given by

\mathbf{r} \left( t \right) = \frac{\sqrt{2}}{2}t\mathbf{i} + e^{t/2}\mathbf{j} + e^{-t/2}\mathbf{k},

where -2 \leq t \leq 5.

 

2)Given that the acceleration vector is \mathbf{a} \left( t \right) = \left( -25 \cos \left( -5 t \right) \right) \mathbf{i} + \left( -25 \sin \left( -5 t \right) \right) \mathbf{j} + \left( -2 t \right) \mathbf{k}, the initial velocity is \mathbf{v} \left( 0 \right) = \mathbf{i + k}, and the initial position vector is \mathbf{r} \left( 0 \right) = \mathbf{i + j + k}, compute:

A. The velocity vector \mathbf{v} \left( t \right) = [removed] \mathbf{i} + [removed] \mathbf{j} + [removed] \mathbf{k}

B. The position vector \mathbf{r} \left( t \right) = [removed] \mathbf{i} + [removed] \mathbf{j} + [removed] \mathbf{k}

Note: the coefficients in your answers must be entered in the form of expressions in the variable \emph{t}; e.g. "5 cos(2t)"

 

3)A gun has a muzzle speed of 50 meters per second. What angle of elevation should be used to hit an object 190 meters away? Neglect air resistance and use g = 9.8 m / sec^{2} as the acceleration of gravity.

 

4)If \mathbf{r} (t) = -4 t^2 \mathbf{i} + 4 t \mathbf{j} + (t^2-5 t)\mathbf{k} gives the position of a particle at time t, find the time t at which the speed of the particle is minimized.

 

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