**2. Projectile motion**

This example studies the effect of small damping on the motion of a particle. Consider a particle of mass *M* which is projected vertically upward with an initial speed *U*0. Let *U* denote the speed at some general time *T*. If air resistance is neglected then the only force acting on the particle is gravity, �*Mg*, where *g* is the acceleration due to gravity and the minus sign occurs because the upward direction is chosen to be the positive direction. Newton's second law governs the motion of the projectile, i.e.,

$$M\frac{dU}{dT} = -M\mathbf{g}.\tag{1}$$

Integrating (1), we obtain the solution *U* ¼ *C* � *gT*. The constant of integration is determined from the initial condition *U*ð Þ¼ 0 *U*0, so that

$$U = U\_0 - \mathbf{g}T.\tag{2}$$

On defining the non-dimensional velocity *v*, and time *t*, by *v* ¼ *U=U*<sup>0</sup> and *t* ¼ *gT=U*0, the governing equation becomes

$$\frac{dv}{dt} = -\mathbf{1}, \quad v(\mathbf{0}) = \mathbf{1}, \tag{3}$$

with the solution *v t*ðÞ¼ 1 � *t*.

Taking account of the air resistance, and is included in the Newton's second law as a force dependent on the velocity in a linear way, we obtain the following linear equation

$$M\frac{dU}{dT} = -M\mathbf{g} - K\mathbf{U},\tag{4}$$

where the drag constant *K* is the dimensions of masa/time. In the nondimensional variables, it becomes

$$\frac{dv}{dt} = -\mathbf{1} - \left(\frac{KU\_0}{M\mathbf{g}}\right)v.\tag{5}$$

Let us denote the dimensionless drag constant by *ε*, then the governing equation is

$$\frac{dv}{dt} = -\mathbf{1} - \epsilon v, \qquad v(0) = \mathbf{1},\tag{6}$$

where *ε*> 0 is a "small" parameter and the disturbances are very "small". The damping constant *K* in (4) is small, since *K* has the dimensions of mass/time and a small quantity in units of kilograms per second.
