Acceleration
The time rate of change of velocity of an object, or alternately the net force on the object divided by the object's mass.
Mathematical Representation
\begin
[ \vec
= \frac{d\vec{v}}
]\end
or
\begin
[ \vec
=\frac{\sum \vec{F}}
]\end
One-Dimensional Acceleration
Utility of the One-Dimensional Case
As with all vector equations, the equations of kinematics are usually approached by separation into components. In this fashion, the equations become three simultaneous one-dimensional equations. Thus, the consideration of motion in one dimension with acceleration can be generalized to the three-dimensional case.
Useful Digrammatic Representations
Several diagrammatic representations are commonly used to represent accelerated motion.
- *Position vs. Time Graph: A plot of position as a function of time is an often useful diagrammatic representation of kinematics problems.
- [*Velocity vs. Time Graph]: The page velocity versus time graph could not be found.
- [Motion Diagram]: A pictorial representation of the motion of an object. it usually takes the form of a one- or two-dimensional plot showing the position of the object at defined times.
Deceleration
In physics, the term acceleration denotes a vector, as does velocity. When the acceleration of an object points in the same direction as its velocity, the object speeds up. When the acceleration of an object points in the direction opposite the object's velocity, the object slows down. In everyday speech, we would call the first case "acceleration" and the second case "deceleration". In physics, both cases represent acceleration, but with a different relationship to the velocity.
Constant Acceleration
Integration with Respect to Time "> Integration with Respect to Time
If acceleration is constant, the definition of acceleration can be integrated:
\begin
[ \int_{v_{\rm i}}^
dv = \int_{t_{\rm i}}^
a\: dt ] \end
For the special case of constant acceleration, the integral yields:
\begin
[ v - v_
= a(t-t_
) ] \end
which is equivalent to:
\begin
[ v = v_
+ a (t-t_
) ] \end
We can now substitute into this equation the definition of velocity,
\begin
[ v = \frac
]\end
which gives:
\begin
[ \frac
= v_
+ a t - a t_
] \end
We can now integrate again:
\begin
[ \int_{x_{\rm i}}^
dx = \int_{t_{\rm i}}^
\left( v_
- at_
+ a t\right)\:dt ] \end
to find:
\begin
[ x - x_
= v_
(t-t_
) - a t_
(t-t_
) + \frac
a( t^
- t_
^
) ] \end
We finish up with some algebra:
\begin
[ x = x_
+ v_
(t-t_
) + \frac
a (t^
- 2 t t_
+ t_
^
) ] \end
which is equivalent to:
\begin
[ x = x_
+ v_
(t-t_
) + \frac
a (t - t_
)^
] \end
Integration with Respect to Position "> Integration with Respect to Position
The definition of acceleration can also be integrated with respect to position, if we use a calculus trick that relies on the chain rule. Returning to the definition of acceleration:
\begin
[ \frac
= a ] \end
we would like to find an expression for v as a function of x instead of t. One way to achieve this is to use the chain rule to write:
\begin
[ \frac
\frac
= a ] \end
We can now elminate t from this expression by using the defnition of velocity to recognize that dx/dt = v. Thus:
\begin
[ \frac
v = a ] \end
which is easily integrated for the case of constant acceleration:
\begin
[ \int_{v_{\rm i}}^
v \:dv = \int_{x_{\rm i}}^
a \:dx ] \end
to give:
\begin
[ v^
= v_
^
+ 2 a (x-x_
) ] \end
One-DimensionalMotionwithConstantAcceleration"> One-Dimensional Motion with Constant Acceleration
Four or Five Useful Equations "> Four or Five Useful Equations
For a time interval during which the acceleration is constant, the instantaneous acceleration at any time will always be equal to the average acceleration. Thus, by analogy with the definition of average velocity, we can write:
\begin
[ a = \langle a\rangle_
= \frac
= \frac{v - v_{\rm i}}{t - t_{\rm i}} ] \end
Taking this equation as a starting point and using the relationship between average velocity and position
\begin
[ \langle v\rangle_
= \frac
= \frac{x - x_{\rm i}}{t- t_{\rm i}} ] \end
We removed the brackets in the acceleration equation above because acceleration is constant. We cannot similarly drop the brackets in the average velocity equation, because velocity is not constant when acceleration is constant (except for the trivial case of a = 0).
lets us derive five very important equations.
Three of these equations follow directly from the integrations performed in the section above.
\begin
[ x = x_
+ \bar
(t-t_
) ] [ \bar
= \frac
(v+v_
) ]
[ v = v_
+ a(t-t_
) ][ x = x_
+ v_
(t-t_
) + \frac
a (t-t_
)^
]
[ v^
= v_
^
+ 2 a (x-x_
) ]\end
Because the first equation is not specific to the case of constant acceleration (it is simply a definition of average velocity) it is combined with the second equation in the summary on the model specification page for one-dimensional motion with constant acceleration.
The Utility of Constant Acceleration">The Utility of Constant Acceleration
Stringing together a series of constant velocity segments is not usually a realistic description of motion, because real objects cannot change their velocity in a discontinuous manner. This drawback does not apply to constant acceleration, however. Objects can have their acceleration changed almost instantaneously. For example, you could be coasting along in a car at a constant 60 mph with zero acceleration when suddenly you see traffic stopped ahead. If you slam on the brakes, your car will still be going 60 mph for an instant, then it will drop to 59, 58, 57, etc. Your acceleration, on the other hand, has almost instantly changed from zero to a substantial acceleration directed opposite your motion. You can feel this abrupt change as a passenger as you are forced against your seatbelt. Similarly, when an airplane begins its takeoff run, you can feel yourself suddenly pressed back in your seat as the plane's acceleration changes almost instantaneously from 0 to a significant forward acceleration. Because of this, it is often reasonable to approximate a complicated motion by separating it into segments of constant acceleration.