If the acceleration of a particle varies in time, its motion can be complex and difficult to analyze. However, a very common and simple type of one-dimensional motion is that in which the acceleration is constant i.e one-dimensional motion with constant acceleration. When this is the case, the average acceleration over any time interval equals the instantaneous acceleration at any instant within the interval, and the velocity changes at the same rate throughout the motion.
If we replace ¯ax by ax in Equation 2.5 and take ti = 0 and tf to be any later time t, we find that
This powerful expression enables us to determine an object’s velocity at any time t if we know the object’s initial velocity and its (constant) acceleration. A velocity–time graph for this constant-acceleration motion is shown in Figure 2.10a. The graph is a straight line, the (constant) slope of which is the acceleration ax ; this is consistent with the fact that ax = dvx/dt is a constant. Note that the slope is positive; this indicates a positive acceleration. If the acceleration were negative, then the slope of the line in Figure 2.10a would be negative.
When the acceleration is constant, the graph of acceleration versus time (Fig. 2.10b) is a straight line having a slope of zero.
Figure 2.10 An object moving along the x axis with constant acceleration ax . (a) The velocity–time graph. (b) The acceleration–time graph. (c) The position–time graph.
Because velocity at constant acceleration varies linearly in time according to Equation 2.8, we can express the average velocity in any time interval as the arithmetic mean of the initial velocity vxi and the final velocity vxf:
Note that this expression for average velocity applies only in situations in which the acceleration is constant.
We can now use Equations 2.1, 2.2, and 2.9 to obtain the displacement of any object as a function of time. Recalling that Δx in Equation 2.2 represents xf - xi , and now using t in place of Δt (because we take ti = 0), we can say
We can obtain another useful expression for displacement at constant acceleration by substituting Equation 2.8 into Equation 2.10:
The position–time graph for motion at constant (positive) acceleration shown in Figure 2.10c is obtained from Equation 2.11. Note that the curve is a parabola. The slope of the tangent line to this curve at t = ti = 0 equals the initial velocity vxi , and the slope of the tangent line at any later time t equals the velocity at that time, vxf.
Figure 2.11 Parts (a), (b), and (c) are vx -t graphs of objects in one-dimensional motion. The possible accelerations of each object as a function of time are shown in scrambled order in (d), (e), and (f).
We can check the validity of Equation 2.11 by moving the xi term to the right hand side of the equation and differentiating the equation with respect to time:
Finally, we can obtain an expression for the final velocity that does not contain a time interval by substituting the value of t from Equation 2.8 into Equation 2.10:
For motion at zero acceleration, we see from Equations 2.8 and 2.11 that
That is, when acceleration is zero, velocity is constant and displacement changes linearly with time.
Equations 2.8 through 2.12 are kinematic expressions that may be used to solve any problem involving one-dimensional motion at constant acceleration. Keep in mind that these relationships were derived from the definitions of velocity and acceleration, together with some simple algebraic manipulations and the requirement that the acceleration be constant.
The four kinematic equations used most often are listed in Table 2.2 for convenience. The choice of which equation you use in a given situation depends on what you know beforehand. Sometimes it is necessary to use two of these equations to solve for two unknowns. For example, suppose initial velocity vxi and acceleration ax are given. You can then find (1) the velocity after an interval t has elapsed, using vxf = vxi + axt, and (2) the displacement after an interval t has elapsed, using xf - xi = vxit + ½ axt2. You should recognize that the quantities that vary during the motion are velocity, displacement, and time.
You will get a great deal of practice in the use of these equations by solving a number of exercises and problems. Many times you will discover that more than one method can be used to obtain a solution. Remember that these equations of kinematics cannot be used in a situation in which the acceleration varies with time. They can be used only when the acceleration is constant.