Constant Acceleration ( Linear Motion) & Free Fall
Motion with constant acceleration
- Motion with constant acceleration or uniformly accelerated motion is
that in which velocity changes at the same rate troughout motion.
- When the acceleration of the moving object is constant its
average acceleration and instantaneous acceleration are equal. Thus from
eq. 5 we have
- Let v0 be the velocity at initial time t=0 and v be the velocity of object at some other instant of time say at t2=t then above eq. 7 becomes
or , v=v0+at (8)
- Graphically this relation is represented in figure 8 given below.
- Thus from the graph it can be seen clearly that velocity v at time t is equal to the velocity v0 at time t=0 plus the change in velocity (at).
- In the same way average velocity can be written as
where x0 is the position of object at time t=0 and vavg is the averag velocity between time t=0 to time t.The above equation then gives
x=x0+vavgt (9)
but for the interval t=0 to t the average velocity is
Now from eq. 8 we find
vavg = v0 + ½(at) (11)
putting this in eq. 9 we find
x = x0 + v0t + ½(at2)
or,
x - x0 = v0t + ½(at2) (12)
this is the position time relation for object having uniformly accelerated motion.
- From eq. 12 it is clear that an object at any time t has
quadratic dependance on time, when it moves with constant acceleration
along a straight line and x-t graph for such motion will be parabolic in natureas shown below.
- Equation 8 and 12 are basic equations for constant acceleration
and these two equations can be combined to get yet another relation for
x , v and a eleminating t so, from 8
putting this value of t in equation 12 and solving it we finally get,
v2 = (v0)2 + 2a ( x - x0 ) (13)
- Thus from equation 13 we see that it is velocity displacement
relation between velocities of object moving with constant acceleration
at time t and t=0 and their corresponding positions at these intervals
of time.
- This relation 13 is helpful when we do not know time t.
- Likewise we can also eliminate the acceleration between equation 8 and 12. Thus from equation 8
putting this value of a in equation 12 and solving it we finally get,
( x - x0 ) = ½ ( v0 + v ) t (14)
- Same way we can also eliminate v0 using equation 8 and 12. Now from equation 8
v0 = v - at putting this value of v0 in equation 12 and solving it we finally get,
( x - x0 ) = vt + ½ ( at2 ) (15)
thus equation 15 does not involve initial velocity v0 - Thus these basic equations 8 and 12 , and derived equations 13, 14, and 15 can be used to solve constant acceleration problems.
Free fall acceleration
- Freely falling motion of any body under the effect of gravity is an example of uniformly accelerated motion.
- Kinematic equation of motion under gravity can be obtained by replacing acceleration 'a' in equations of motion by acceleration due to gravity 'g'.
- Value of g is equal to 9.8 m.s-2.
- Thus kinematic equations of motion under gravity are
v = v0 + gt (16a)
x = v0t + ½ ( gt2 ) (16b)
v2 = (v0)2 + 2gx (16c)
- The value of g is taken positive when the body falls vertically downeards and negative when the body is projected up against gravity.
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