## Chapter 6

## 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.

## Step-by-Step

## 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 .