M1.3 Acceleration in Kinematics
Motion with constant velocity is a rather ideal case of motion studied in Physics. If we remember that velocity, being a vector, has value and direction, then it is obvious that most of the motions we can observe in everyday life do not have constant velocity. If the velocity of a moving object is changing it means there is acceleration.
Acceleration: is a rate at which a velocity is changing.
It is usually
denoted by letter a. The change of velocity is a vector so we write the
symbol representing it with an arrow: 
. Acceleration is defined by
the equation
(M1.11)where 
is a
change of velocity which occurs during a time 
. This definition
of acceleration is valid for motion with constant acceleration. It is
analogical to the definition of velocity given by Equation M1.3. An example of
approximately such a motion is a train when it starts from the station. For the
first 50 or 60 seconds it accelerates more or less uniformly. Then it moves
with constant velocity up to the moment when it starts slowing down to stop at
a station.
When the train starts, the acceleration is positive, it has the same direction as
velocity. When the train is slowing down it has negative acceleration with the
direction opposite to the velocity. Such a method of correlating the sign of
acceleration with the direction of the velocity is valid for motion along a
straight line. In a general case acceleration and velocity can have quite
arbitrary relative directions. For example, can be
perpendicular to 
. But
in all cases the change of the sign of acceleration means that it changes its
direction to an opposite one.
To define acceleration for any type of motion we must follow the reasoning described for velocity. Writing acceleration as
(M1.12)
and with a time difference approaching a limiting value of zero we arrive at the most general definition of acceleration
(M1.13)Combining Equations M1.10 and M1.13 we get
(M1.14)This definition of acceleration is valid for any type of motion as is the definition of velocity given by Formula M1.10. Acceleration, as defined by Formula M1.14 is a second derivative of the position of a moving object with respect to time. Dimensions of acceleration follow from Formula M1.11 and are: dimensions of velocity per dimensions of time.
(M1.15)The square brackets [ ] enclosing the symbol of the quantity are customarily used to denote dimensions of this quantity. Notice that there is no arrow above the symbol of acceleration in Formula M1.15. It is because we are talking about dimensions of the acceleration, not the acceleration itself.
To get a “feeling” of what values of acceleration we witness every day, let us consider some examples of motion with constant acceleration.
An object allowed to fall freely downwards moves with acceleration of about 9.8 m/s2. This acceleration is due to gravity.
The acceleration of a typical car that goes from 0 to 100 km/h in 12 seconds is
We expressed this acceleration in SI units to compare it directly to acceleration due to gravity, g = 9.8 m/s2. So, the acceleration of such a car is about 0.24 g.
Some of the best commercially available cars have this parameter equal to 0.7g. What is the time needed for such a car to reach the speed of 100 km/h?
On the basis of Equation M1.11 we can write
0.7 g = ∆v/∆t
so
∆t = ∆v/ 0.7 g
Later on we will learn what a tremendous difference there must be between a car which can accelerate to 100 km/h during 12.0 s and that which requires only 4.0 s for the same result. Knowledge of Physics enable us to understand these differences.