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Motion with constant acceleration
Motion with constant acceleration occurs in everyday life whenever an
object is dropped: the object moves downward with the constant
acceleration
, under the influence of gravity.
Fig. 8 shows the graphs of displacement versus time and
velocity versus time for a body
moving with constant acceleration. It can be seen that the displacement-time graph consists of
a curved-line whose gradient (slope) is increasing in time.
This line can be represented algebraically as
![\begin{displaymath}
x = x_0 + v_0 t + \frac{1}{2} a t^2.
\end{displaymath}](img147.png) |
(19) |
Here,
is the displacement at time
: this quantity can be determined from the
graph as the intercept of the
curved-line with the
-axis. Likewise,
is the body's instantaneous
velocity at time
.
Figure 8:
Graphs of displacement versus time and velocity versus time
for a body moving with constant acceleration
![\begin{figure}
\epsfysize =4.in
\centerline{\epsffile{consta.eps}}
\end{figure}](img149.png) |
The velocity-time graph consists of a straight-line which can be represented
algebraically as
![\begin{displaymath}
v = \frac{dx}{dt}= v_0 + a t.
\end{displaymath}](img150.png) |
(20) |
The quantity
is determined from the graph as the intercept of the
straight-line with the
-axis. The quantity
is the constant acceleration: this
can be determined graphically as the gradient of the straight-line
(i.e., the ratio
, as shown). Note that
, as expected.
Equations (19) and (20) can be rearranged to give the
following set of three useful formulae which characterize motion with
constant acceleration:
Here,
is the net distance traveled after
seconds.
Fig. 9 shows a displacement versus time graph for a slightly more
complicated case of accelerated motion. The body in question accelerates
to the right [since the gradient (slope) of the graph is increasing in time]
between times
and
. The body then moves to
the right (since
is increasing in time) with a constant velocity
(since the graph is a straight line) between times
and
.
Finally, the body decelerates [since the gradient (slope) of the graph is decreasing in time]
between times
and
.
Figure 9:
Graph of displacement versus time
![\begin{figure}
\epsfysize =2in
\centerline{\epsffile{consta1.eps}}
\end{figure}](img160.png) |
Next: Free-fall under gravity
Up: Motion in 1 dimension
Previous: Motion with constant velocity
Richard Fitzpatrick
2006-02-02