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).
(a) Velocity-versus-time graph with constant acceleration showing the initial and final velocities
. The average velocity is
. (b) Velocity-versus-time graph with an acceleration that changes with time. The average velocity is not given by
, but is greater than 60 km/h.
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
and
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:
.
Second, we identify the unknown; in this case, it is final velocity
.
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] ,
.
Solution
Substitute the known values and solve:
[link] is a sketch that shows the acceleration and velocity vectors.
The airplane lands with an initial velocity of 70.0 m/s and slows to a final velocity of 10.0 m/s before heading for the terminal. Note the acceleration is negative because its direction is opposite to its velocity, which is positive.
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.
In addition to being useful in problem solving, the equation
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 =
v0 ), 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
to each side of this equation and dividing by 2 gives
Since
for constant acceleration, we have
Now we substitute this expression for
into the equation for displacement,
, yielding
Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc