3.4 Motion with constant acceleration  (Page 2/10)

In part (b), acceleration is not constant. During the 1-h interval, velocity is closer to 80 km/h than 40 km/h. Thus, the average velocity is greater than in part (a).

Solving for final velocity from acceleration and time

We can derive another useful equation by manipulating the definition of acceleration:

Substituting the simplified notation for $\text{Δ}v$ and $\text{Δ}t$ gives us

Solving for v yields

Calculating final velocity

An airplane lands with an initial velocity of 70.0 m/s and then decelerates at 1.50 m/s ² for 40.0 s. What is its final velocity?

Strategy

First, we identify the knowns: $v_0=70\text{m/s,}a=\mathrm{-1.50}{\text{m/s}}^2,t=40\text{s}$ .

Second, we identify the unknown; in this case, it is final velocity $v_{\text{f}}$ .

Last, we determine which equation to use. To do this we figure out which kinematic equation gives the unknown in terms of the knowns. We calculate the final velocity using [link] , $v=v_0+at$ .

Solution

Substitute the known values and solve:

$v=v_0+at=70.0\text{m/s}+\left(\mathrm{-1.50}\text{m/}{\text{s}}^2\right)\left(\mathrm{40.0\; s}\right)=\mathrm{10.0\; m/s.}$

[link] is a sketch that shows the acceleration and velocity vectors.

Significance

The final velocity is much less than the initial velocity, as desired when slowing down, but is still positive (see figure). With jet engines, reverse thrust can be maintained long enough to stop the plane and start moving it backward, which is indicated by a negative final velocity, but is not the case here.

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In addition to being useful in problem solving, the equation $v=v_0+at$ gives us insight into the relationships among velocity, acceleration, and time. We can see, for example, that

• Final velocity depends on how large the acceleration is and how long it lasts
• If the acceleration is zero, then the final velocity equals the initial velocity ( v = v ₀ ), as expected (in other words, velocity is constant)
• If a is negative, then the final velocity is less than the initial velocity

All these observations fit our intuition. Note that it is always useful to examine basic equations in light of our intuition and experience to check that they do indeed describe nature accurately.

Solving for final position with constant acceleration

We can combine the previous equations to find a third equation that allows us to calculate the final position of an object experiencing constant acceleration. We start with

Adding $v_0$ to each side of this equation and dividing by 2 gives

Since $\frac{v_0+v}{2}=\stackrel{\text{–}}{v}$ for constant acceleration, we have

Now we substitute this expression for $\stackrel{\text{–}}{v}$ into the equation for displacement, $x=x_0+\stackrel{\text{–}}{v}t$ , yielding