Differential Equations and Linear Algebra, 4th Edition

2.3 Acceleration–Velocity Models

In Section 1.2 we discussed vertical motion of a mass m near the surface of the earth under the influence of constant gravitational acceleration. If we neglect any effects of air resistance, then Newton’s second law $(F=ma$) implies that the velocity v of the mass m satisfies the equation

$$m{\displaystyle \frac{dv}{dt}}=F_G,$$ (1)

where $F_G=-mg$ is the (downward-directed) force of gravity, where the gravitational acceleration is $g\approx 9.8{\text{m/s}}^2$ (in mks units; $g\approx 32{\text{ft/s}}^2$ in fps units).

Example 1

No air resistance Suppose that a crossbow bolt is shot straight upward from the ground ($y_0=0$) with initial velocity $v_0=49$ (m/s). Then Eq. (1) with $g=9.8$ gives

$${\displaystyle \frac{dv}{dt}}=-9.8,\text{so}v(t)=-(9.8)t+v_0=-(9.8)t+49.$$

Hence the bolt’s height function y(t) is given by

$$y(t)$$