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## Q. 6.12

Projectile motion with air resistance. A projectile S of mass m is fired from a gun with muzzle speed  $v_0$  at an angle β with the horizontal plane. Neglect the Earth’s motion and wind effects and assume that air resistance is govern ed by Stokes’s law. Determine the projectile’s motion as a function of time.

## Verified Solution

The equations of motion with air resistance governed by Stokes’s law are given in (6.36) . To find the motion x(S, t) , we first integrate the system (6.36) to obtain v(S, t ). Use of the initial condition  $\mathbf{v}_0=v_0(\cos \beta \mathbf{i} + \sin \beta \mathbf{j})$  yields

$\ddot{x}=-v \dot{x}, \quad \ddot{y}=-g – v \dot{y} \quad \text { with } \quad v \equiv \frac{c}{m} .$              (6.36)

$\int_{v_0 \cos \beta}^{\dot{x}} \frac{d \dot{x}}{\dot{x}}=-v t, \quad \int_{v_0 \sin \beta}^{\dot{y}} \frac{d \dot{y}}{g + v \dot{y}}=-t .$

These deliver the projectile’s velocity components as functions of time:

$\dot{x}=\left(v_0 \cos \beta\right) e^{-v t}, \quad \dot{y}=-\frac{g}{v} + \left(v_0 \sin \beta + \frac{g}{v}\right) e^{-v t}$.          (6.37a)

Then integration of (6.37a) with the initial condition  $\mathbf{x}_0=\mathbf{0}$  yields the motion of the projectile as a function of time:

$x(t)=\frac{v_0 \cos \beta}{v}\left(1 – e^{-v t}\right)$                 (6.37b)

$y(t)=-\frac{g}{v} t + \frac{1}{v}\left(v_0 \sin \beta + \frac{g}{v}\right)\left(1 – e^{-v t}\right)$             (6.37c)

Let us imagine that the projectile is fired from a hilltop into a wide ravine, as shown in Fig. 6.10. Then, as  $t \rightarrow \infty$,  in the absence of impact, (6.37a) gives  $\dot{x} \rightarrow 0$  and  $\dot{y} \rightarrow-g / \nu$.  Hence, the projectile attains the terminal speed  $v_{\infty}=g / \nu$   at which its weight is balanced by air  resistance; and (6.37b) and (6.37c) show that the projectile approaches asymptotically, the vertical range line at  $r_{\infty} \equiv \lim _{t \rightarrow \infty} x(t)=\left(v_0 \cos \beta\right) / v$  in Fig. 6.10. In the absence of air resistance, the range for the same situation would grow indefinitely with the width of the ravine. The simple Stokes model thus provides a more realistic picture of projectile motion with air resistance that limits its range .