IIT (NTSE/Olympiad)  

Motion in one Dimension

Motion With Uniform Acceleration

Let = Initial velocity (at t = 0), = Velocity of the particle after time t, = Acceleration (uniform), = Displacement of the particle during time 't'
(a) Acceleration, [Because of uniform acceleration, this acceleration is instantaneous as well average acceleration]. From above equation = + t ....(i)
(b) Displacement = Average velocity × time, ....(ii)
[This is very useful equation, when acceleration is not given]
(c) From (i) and (ii) = t + (1/2) t2 ....(iii)
Again from (i) and (iii)
= t − (1/2) t2.
[Here negative sign does not indicate that retardation is occurring]
(d) From (i) and (ii)2 = 2 + 2 .....(iv)

[This equation is dimensionally non balanced because we have substituted value of t = 1s and second is neglected that's why it seems to be unbalanced]
Equations (i), (iii) and (iv) one called 'equations of motion' and are very useful in solving the problems of motion along a straight line with constant acceleration.
Note :
(i) These equations can be applied only and only when acceleration is constant. In case of circular motion or simple harmonic motion as acceleration is not constant (due to change in direction or magnitude) so these equation can not be applied.
(ii) = + t and = t + (1/2)t2 are vector equation, while
= + 2
⇒ v2 = u2 + 2 is a scalar equation
(iii) If the velocity and acceleration are collinear, we conventionally take the direction of motion to be positive, so equation of motions becomes v = u + at, s = ut + (1/2) at2, v2= u2+ 2as
If the velocity and acceleration are anti-parallel then, v = u – at, s = ut – (1/2) at2 and v2 = u2 – 2as

IIT (Class X)