2.4.2 The Kinematic Equations of Constant Acceleration

Consider an object whose acceleration $a_{s}$ remains constant during the time interval $Δ t = t_{f} - t_{i}$ . At the beginning of this interval, at time $t_{i}$ , the object has initial velocity $v_{i s}$ and initial position $s_{i}$ . Note that $t_{i}$ is often zero, but it does not have to be. We would like to predict the object’s final position $s_{f}$ and final velocity $v_{f s}$ at time $t_{f}$ .

The object’s velocity is changing because the object is accelerating. FIGURE 2.22a shows the acceleration-versus-time graph, a horizontal line between $t_{i}$ and $t_{f}$ . It is not hard to find the object’s velocity $v_{f s}$ at a later time $t_{f}$ . By definition,

The figure consists of two parts labeled ("a") and (b) that depict an acceleration-versus-time graph and a velocity-versus-time graph of an object with constant acceleration.

FIGURE 2.22 Full Alternative Text

a_{s} = \frac{Δ v_{s}}{Δ t} = \frac{v_{f s} - v_{i s}}{Δ t}

(2.18)

which is easily rearranged to give

v_{f s} = v_{i s} + a_{s} Δ t

(2.19)

The velocity-versus-time graph, shown in FIGURE 2.22b, is a straight line that starts at $v_{i s}$ and has slope $a_{s}$ .

A photo of an integrated circuit chip with an accelerometer.

A schematic diagram of an accelerometer.

Full Alternative Text

As you learned in the last section, the object’s final position is

s_{f} = s_{i} + area under the velocity curve v_{s} {between t}_{i} and t_{f}

(2.20)

The shaded area in Figure 2.22b can be subdivided into a rectangle of area $v_{i s} Δ t$ and a triangle of area $\frac{1}{2} (a_{s} Δ t) (Δ t) = \frac{1}{2} a_{s} {(Δ t)}^{2}$ . Adding these gives

s_{f} = s_{i} + v_{i s} Δ t + \frac{1}{2} a_{s} {(Δ t)}^{2}

(2.21)

where $Δ t = t_{f} - t_{i}$ is the elapsed time. The quadratic dependence on $Δ t$ causes the position-versus-time graph for constant-acceleration motion to have a parabolic shape, as shown in Model 2.2.

Equations 2.19 and 2.21 are two of the basic kinematic equations for motion with constant acceleration. They allow us to predict an object’s position and velocity at a future instant of time. We need one more equation to complete our set, a direct relation between position and velocity. First use Equation 2.19 to write $Δ t = (v_{f s} - v_{i s}) / a_{s}$ . Substitute this into Equation 2.21, giving

s_{f} = s_{i} + v_{i s} (\frac{v_{f s} - v_{i s}}{a_{s}}) + \frac{1}{2} a_{s} (\frac{v_{f s} - v_{i s}}{a_{s}})^{2}

(2.22)

With a bit of algebra, this is rearranged to read

v_{f s}^{2} = v_{i s}^{2} + 2 a_{s} Δ s

(2.23)

where $Δ s = s_{f} - s_{i}$ is the displacement (not the distance!). Equation 2.23 is the last of the three kinematic equations for motion with constant acceleration.