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[Mechanics] Projectile Motion with Stokes' Flow

2021-10-11 08:48 作者:AoiSTZ23 0人讀過 | 我要投稿

By: Tao Steven Zheng (鄭濤)

【Problem】

Assume the air resistance is proportional to the projectile's velocity. Determine the trajectory of a projectile fired at origin, and with initial velocity

$%5Cbegin%7Bpmatrix%7D%20v_x%20%5C%5C%20v_y%20%5C%5C%20v_z%20%5Cend%7Bpmatrix%7D%5C%20%20%3D%20%5Cbegin%7Bpmatrix%7D%20%5Cdot%20x_0%20%5C%5C%20%5Cdot%20y_0%20%5C%5C%20%5Cdot%20z_0%20%5Cend%7Bpmatrix%7D%5C$

(1) Determine the displacement vector.

(2) Let there be no motion in the y-axis. Determine $%20z(x)$ .

【Solution】

Part (1)
To model the forces of air resistance, consider the net force in three dimensions

$%5Cbegin%7Bpmatrix%7D%20ma_x%20%5C%5C%20ma_y%20%5C%5C%20ma_z%20%5Cend%7Bpmatrix%7D%5C%20%20%3D%20%5Cbegin%7Bpmatrix%7D%20-kv_x%20%5C%5C%20-kv_y%20%5C%5C%20-kv_z%20-%20mg%20%5Cend%7Bpmatrix%7D%5C$

Divide the mass on both sides of the equation. Define $%5Crho%20%3D%20%5Cfrac%7Bk%7D%7Bm%7D$ , and rewrite the vector quantities in terms of the coordinates $x%2C%20y%2C%20z%20$ :

$%5Cbegin%7Bpmatrix%7D%20%5Cddot%20x%20%5C%5C%20%5Cddot%20y%20%5C%5C%20%5Cddot%20z%20%5Cend%7Bpmatrix%7D%5C%20%20%3D%20%5Cbegin%7Bpmatrix%7D%20-%5Crho%20%5Cdot%20x%20%5C%5C%20-%5Crho%20%5Cdot%20y%5C%5C%20-%5Crho%20%5Cdot%20z%20-%20g%20%5Cend%7Bpmatrix%7D%5C$

To obtain the velocity vector, integrate the acceleration with respect to time. Use the initial condition to solve for the constants.

$%5Cbegin%7Bpmatrix%7D%20%5Cdot%20x%20%5C%5C%20%5Cdot%20y%20%5C%5C%20%5Cdot%20z%20%5Cend%7Bpmatrix%7D%5C%20%3D%20%5Cbegin%7Bpmatrix%7D%0A%5Cdot%20x_0%20%7Be%7D%5E%7B-%5Crho%20t%7D%20%5C%5C%0A%5Cdot%20y_0%20%7Be%7D%5E%7B-%5Crho%20t%7D%20%5C%5C%0A%5Cfrac%7B-g%7D%7B%5Crho%7D%20%2B%20%5Cleft(%5Cdot%20z_0%20%2B%20%5Cfrac%7Bg%7D%7B%5Crho%7D%20%5Cright)%20%7Be%7D%5E%7B-%5Crho%20t%7D%0A%5Cend%7Bpmatrix%7D%5C$

To obtain the displacement vector, again, integrate the velocity with respect to time. Use the initial condition to solve for the constants.

$%5Cbegin%7Bpmatrix%7D%20x%20%5C%5C%20y%20%5C%5C%20z%20%5Cend%7Bpmatrix%7D%5C%20%3D%20%5Cbegin%7Bpmatrix%7D%0A%5Cfrac%7B1%7D%7B%5Crho%7D%5Cdot%20x_0%20%5Cleft(1%20-%20%7Be%7D%5E%7B-%5Crho%20t%7D%20%5Cright)%20%5C%5C%0A%5Cfrac%7B1%7D%7B%5Crho%7D%5Cdot%20y_0%20%5Cleft(1%20-%20%7Be%7D%5E%7B-%5Crho%20t%7D%20%5Cright)%20%5C%5C%0A%5Cfrac%7B-g%7D%7B%5Crho%7D%20t%20%2B%20%5Cfrac%7B1%7D%7B%5Crho%7D%20%5Cleft(%5Cdot%20z_0%20%2B%20%5Cfrac%7Bg%7D%7B%5Crho%7D%20%5Cright)%20%5Cleft(1%20-%20%7Be%7D%5E%7B-%5Crho%20t%7D%20%5Cright)%0A%5Cend%7Bpmatrix%7D%5C%20$

Part (2)
Since $x%20%3D%20%5Cfrac%7B1%7D%7B%5Crho%7D%5Cdot%20x_0%20%5Cleft(1%20-%20%7Be%7D%5E%7B-%5Crho%20t%7D%20%5Cright)$ , it follows that

$%20t%20%3D%20%5Cfrac%7B1%7D%7B%5Crho%7D%20%5Cln%20%5Cleft(%5Cfrac%7B%5Cdot%20x_0%7D%7B%5Cdot%20x_0%20-%20%5Crho%20x%7D%20%5Cright)$

Therefore,

$z(x)%20%3D%20%5Cfrac%7B-g%7D%7B%7B%5Crho%7D%5E%7B2%7D%7D%5Cln%20%5Cleft(%5Cfrac%7B%5Cdot%20x_0%7D%7B%5Cdot%20x_0%20-%20%5Crho%20x%7D%20%5Cright)%0A%2B%20%5Cleft(%5Cdot%20z_0%20%2B%20%5Cfrac%7Bg%7D%7B%5Crho%7D%20%5Cright)%5Cfrac%7Bx%7D%7B%5Cdot%20x_0%7D$

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