## A ball is thrown in the wind

A ball of mass $m$ is thrown and experiences a frictional drag force as it moves through a gas or a fluid. The forces acting on this ball are gravity $-mg\hat{z}$ and the drag force. If a wind is blowing, the drag force can be described by,

$\vec{F}_{fric} = -a(\vec{v}-\vec{v}_{\text{wind}}) - b(\vec{v}-\vec{v}_{\text{wind}})|(\vec{v}-\vec{v}_{\text{wind}})|,$

where $a$ and $b$ are constants and $\vec{v}_{\text{wind}}$ is the velocity of the wind which can depend on position and time. For low Reynolds number, the linear term $-a(\vec{v}-\vec{v}_{\text{wind}})$ usually dominates whereas for high Reynolds number, the quadratic term $- b(\vec{v}-\vec{v}_{\text{wind}})|(\vec{v}-\vec{v}_{\text{wind}})|$ dominates.

$\vec{F}= m\frac{d^2\vec{r}}{dt^2} = -a(\vec{v}-\vec{v}_{\text{wind}}) - b(\vec{v}-\vec{v}_{\text{wind}})|(\vec{v}-\vec{v}_{\text{wind}})|-mg\,\hat{z}$

$m=$  kg $a=$  N s/m $b=$  N s²/m²

The three components of the wind vectors can be functions of space and time.
$v_{\text{wind},x}=$  m/s $v_{\text{wind},y}=$  m/s $v_{\text{wind},z}=$  m/s

The initial conditions at time $t=0$ are,
$x=$  m  $y=$  m  $z=$  m  $v_x=$  m/s  $v_y=$  m/s  $v_z=$  m/s

### 3-D motion differential equation solver

 $F_x=$ -x [N] $F_y=$ 0 [N] $F_z=$ 0 [N] $m=$ [kg] Initial conditions: $t_0=$ [s] $\Delta t=$ [s] $x(t_0)=$ [m] $N_{steps}$ $v_x(t_0)=$ [m/s] Plot: t x vx y vy z vz Fx Fy Fz P Ekin W vs. t x vx y vy z vz Fx Fy Fz P Ekin W $y(t_0)=$ [m] $v_y(t_0)=$ [m/s] $z(t_0)=$ [m] $v_z(t_0)=$ [m/s]

= , =

 the animation to zoom or rotate.

 $t$ [s] $x$ [m] $y$ [m] $z$ [m] $v_x$ [m/s] $v_y$ [m/s] $v_z$ [m/s] $F_x$ [N] $F_y$ [N] $F_z$ [N] $P$ [W] $E_{\text{kin}}$ [J] $W$ [J]