# Problem: A ball is shot from a compressed air gun at twice its terminal speed.a)What is the ball's initial acceleration, as a multiple of g, if it is shot straight up?b) What is the ball's initial acceleration, as a multiple of g, if it is shot straight down?

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###### FREE Expert Solution

Drag force on the ball is given by:

$\overline{){\mathbf{D}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{C}}{\mathbf{\rho }}{\mathbf{A}}{{\mathbf{v}}}^{{\mathbf{2}}}}$

We'll also use Newton's second law:

$\overline{){\mathbf{\Sigma F}}{\mathbf{=}}{\mathbf{m}}{\mathbf{a}}}$

This implies that:

D = ma = mg

The velocity of the ball is twice its terminal velocity.

$\begin{array}{rcl}{\mathbf{D}}_{\mathbf{2}{\mathbf{v}}_{\mathbf{\tau }}}& \mathbf{=}& \frac{\mathbf{1}}{\mathbf{2}}\mathbf{C}\mathbf{\rho }\mathbf{A}\mathbf{\left(}\mathbf{2}{\mathbf{v}}^{\mathbf{2}}\mathbf{\right)}\\ & \mathbf{=}& \mathbf{4}\mathbf{\left(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{C}\mathbf{\rho }\mathbf{A}{\mathbf{v}}^{\mathbf{2}}\mathbf{\right)}\end{array}$

D2v = 4D = 4mg ###### Problem Details

A ball is shot from a compressed air gun at twice its terminal speed.

a)What is the ball's initial acceleration, as a multiple of g, if it is shot straight up?
b) What is the ball's initial acceleration, as a multiple of g, if it is shot straight down?