4.1 Basics of differential equations  (Page 4/14)

In physics and engineering applications, we often consider the forces acting upon an object, and use this information to understand the resulting motion that may occur. For example, if we start with an object at Earth’s surface, the primary force acting upon that object is gravity. Physicists and engineers can use this information, along with Newton’s second law of motion (in equation form $F=ma,$ where $F$ represents force, $m$ represents mass, and $a$ represents acceleration), to derive an equation that can be solved.

In [link] we assume that the only force acting on a baseball is the force of gravity. This assumption ignores air resistance. (The force due to air resistance is considered in a later discussion.) The acceleration due to gravity at Earth’s surface, $g,$ is approximately $9.8{\text{m/s}}^2.$ We introduce a frame of reference, where Earth’s surface is at a height of 0 meters. Let $v\left(t\right)$ represent the velocity of the object in meters per second. If $v\left(t\right)>0,$ the ball is rising, and if $v\left(t\right)<0,$ the ball is falling ( [link] ).

Our goal is to solve for the velocity $v(t)$ at any time $t.$ To do this, we set up an initial-value problem. Suppose the mass of the ball is $m,$ where $m$ is measured in kilograms. We use Newton’s second law, which states that the force acting on an object is equal to its mass times its acceleration $(F=ma).$ Acceleration is the derivative of velocity, so $a(t)=v^{\prime }(t).$ Therefore the force acting on the baseball is given by $F=m{v}^{\prime }(t).$ However, this force must be equal to the force of gravity acting on the object, which (again using Newton’s second law) is given by $F_g=\text{−}mg,$ since this force acts in a downward direction. Therefore we obtain the equation $F=F_g,$ which becomes $m{v}^{\prime }(t)=\text{−}mg.$ Dividing both sides of the equation by $m$ gives the equation

Notice that this differential equation remains the same regardless of the mass of the object.

We now need an initial value. Because we are solving for velocity, it makes sense in the context of the problem to assume that we know the initial velocity    , or the velocity at time $t=0.$ This is denoted by $v(0)=v_0.$