Motion with constant acceleration

Active vocabulary

uniformly uniformly accelerated motion uniformly decelerated motion

——

рівномірно прискорений рух рівномірно сповільнений рух

In many common types of motion, the acceleration is either constant or approximately so (Fig. 1.8). Such cases are so ubiquitous that a special set of equations was derived for dealing with them.

When the acceleration is constant, the average acceleration and instantaneous acceleration are equal and we can write:

Where is the velocity at time t = 0 and is the velocity at any later time. We can recast this equation as

Fig. 1.8

For a check, note that this equation is reduced to for t = 0 as it must be. For next check, take the derivative of the last equation. Doing so yields du/dt = a, which is the definition of a. Figure 1.8 b shows a plot of this equation, the u(t) function.

In a similar manner we can write

and then as

in which is the position of the particle at t = 0, and is the average velocity between t = 0 and a later time t

If you plot u against t, a straight line results. Under these conditions, the averagevelocity over any time interval (say, t = 0 to a later time t) is the average of the velocity at the beginning of the interval (= ) and the velocity at the end of the interval (= ). From the interval t = 0 to the later time t,then, the average velocity is

Substituting the right side of equation for yields, after a little rearrangement,

Finally, substituting this equation into equation for x yields

For a check, note that putting t = 0 yields x = x₀, as it must be. For next check, taking the derivative of the last equation we get equation for velocity u, again as it must be.

Equations for velocity and path can also be combined in three ways to yield three additional equations. First, we can eliminate t to obtain

This equation is useful if we do not know t and do not require to find it. Second, we can eliminate the acceleration a:

Finally, we can eliminate , obtaining

The first two equations in Table 1 are the basic equations from which the others are derived. Those two can be obtained by integrating of the acceleration under the condition that a is constant. The definition of a is

,

which can be rewritten as

To take the indefinite integral(or antiderivative) of both sides, we write

Table 1.1 gives equations which are valid only for constant acceleration and shows which one of the five possible quantities is missing from each.

Table 1.1

equation	missing quantity

Since acceleration a is constant, it can be taken outside the integration. Then we obtain

Fig. 1.9

or

To evaluate the constant of integration C,we let t = 0, at which time Insert these values into previous equation (which must hold for all values of t, including t = 0) yields

Finally we get

To derive the other basic equation in Table 1, we rewrite the definition of velocity as (fig. 1.9).

and then take the indefinite integral of both sides to obtain

There is no reason to believe that is constant, so we cannot move it outside the integration. But we can write:

Since is constant, this can be rewritten as

Integration now yields

where C' is another constant of integration. At time t = 0, we have x = x₀. Substitution of these values in latter gives x₀ = C'. Replacing C' with x₀ gives us

---

Дата добавления: 2015-11-16