If u is the
initial velocity of a moving body, and if the velocity of the body changes
to v (which we shall call the final velocity) in a time interval t, then
the acceleration of the body in the time t is given by
Note that in the
above formula we assume that the acceleration of the body was uniform
(i.e. the same) throughout the time interval t. Below is a car moving with
a uniform acceleration.
Now, equation
(0) can be re-written as
at =
v-u
=> v-u =
at
average
velocity = (u+v)/2
Now, the distance
s, traveled in the time t by the body is given by
distance
traveled = average velocity x time
s = [(u+v)/2]t
From equation
(1) we have v=u+at, substituting this in the above equation for v, we
get
s = [(u+u+at)/2]t
=> s =
[(2u+at)/2]t
=> s = [(u
+ (1/2)at)]t
This is Newton's
second equation of Motion. This equation can be used to calculate the
distance traveled by a body moving with a uniform acceleration in a time
t. Again here, if the body started from rest, then we shall
substitute u=0 in this equation.
If you take a
close look at the 2 equations of motion we derived just now you can
observe that none of these equations carry a relation between distance
traveled and final velocity of the body. All other relations are
available. So, there is a need to find an equation which relates s and v.
We derive it as follows.
We start with
squaring equation (1). Thus we have
v2
= (u+at)2
=>
v2 = u2
+ a2t2
+ 2uat
=>
v2 = u2
+ 2uat + a2t2
=>
v2 = u2
+ 2a(ut + (1/2)at2)
now, using equation
2 we have
=>
v2 = u2
+
2as
- (3)
As you can see,
the above equation gives a relation between the final velocity v of the
body and the distance s traveled by the body.
Thus, we have the
the three Newton's equations of Motion as
1) v= u + at
2) s = ut + (1/2)at2
3) v2 = u2
+ 2as
No comments:
Post a Comment
Your Comment is posted.