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 v_avg is the averag velocity between time t=0 to time t.The above equation then gives
  x=x₀+v_avgt                    (9)
  but for the interval t=0 to t the average velocity is
  
  
  Now from eq. 8 we find
  v_avg = 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.