Acceleration and Equations of Motion

Acceleration and Equations  of Motion

ACCELERATION & EQUATIONS OF MOTION
If the velocity of a particle is changing as it moves then it is said to be moving with acceleration. The acceleration measures how rapidly the velocity is changing.
Acceleration is defined as the rate of change of velocity.
               \( \therefore a = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta V}} {{\Delta t}} = \frac{{dV}} {{dt}} \)

The average acceleration is defined as the ratio of change in velocity over a time interval to the time interval.
Let v₁,v₂ be the velocities of a particle at instants t₁,t₂ respectively. Now,
    Average acceleration= \( \frac{{\text{change in velocity}}} {{\text{time}}} \)
            \( \therefore a = \frac{{V_2 - V_1 }} {{t_2 - t_1 }} = \frac{{\Delta V}} {{\Delta t}} \)
It is a vector. It is in the direction of change in velocity.
S.I. Unit is ms^-2, dimensional formula is [\( L^1 T^{ - 2} \)]

Note : The velocity variation may be due to change in magnitude of velocity (speed) or change in direction of velocity. Hence acceleration may be due to either of the above reasons or both.

Note : The acceleration of a moving particle may be positive or negative. If the speed of particle is increasing with time then acceleration is positive and if the speed is decreasing with time then acceleration is negative.This statement is independent of the choice of reference axis

Note : For positive acceleration the velocity vector and acceleration vector are in the same direction. But for negative acceleration, the velocity and acceleration vectors are opposite.

Acceleration and Equations of Motion

INSTANTANEOUS ACCELERATION

The acceleration of a particle at a particular instant of time is called it's instantaneous acceleration.
It is also defined as the limit of average acceleration as the time interval (At) becomes infinitesimally small.
If the time interval \(\Delta t\) is chosen to be very small, i.e., as \(\Delta t\to0\), the corresponding acceleration is called instantaneous acceleration.
       \( \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta V}} {{\Delta t}} = \frac{{dV}} {{dt}} \) =Instantaneous acceleration.

        \( a = \frac{{dv}} {{ds}}.\frac{{ds}} {{dt}};a = v.\frac{{dv}} {{ds}} \)    a.ds=v.dv

Acceleration and Equations of Motion

DECELERATION OR RETARDATION

If the speed is decreasing with time then acceleration is negative.
The negative acceleration is called deceleration or retardation.
 

Acceleration and Equations of Motion

UNIFORM ACCELERATION

If the average acceleration over any time interval equals the instantaneous acceleration at any instant of time then the acceleration is said to be uniform or constant. It does not vary with time. 
The velocity either increases or decreases at the same rate throughout the motion.                     
                                (or)
If a body has equal changes in velocities in equal intervals of time however small the intervals may be, then it is set to move it uniform acceleration.