# Average Acceleration , Instantaneous Acceleration

### (a) Average Acceleration

If the velocity (magnitude, direction or both) of a particle changes with time, its motion is said to be non-uniform.

Suppose that the velocity of particle changes by Δv over a time interval Δt, the time rate of change of velocity is given by Δv/Δt, this is known as the average acceleration of the particle over the time interval Δt

$\displaystyle \vec{a_{av}} = \frac{\vec{\Delta v}}{\Delta t}$

### (b) Instantaneous Acceleration

For an instant (infinitely small or infinitesimal time interval), the change in velocity of the particle is infinitely small, but the ratio of infinitesimal change in velocity and the infinitesimal time is finite. This finite ratio is known as instantaneous acceleration:

$\displaystyle \vec{a} = \frac{\vec{d v}}{d t}$

### (c) Uniform Acceleration

When a particle undergoes constant acceleration (vector) for some time interval we say that it is moving with constant or uniform acceleration over that time interval.

#### Average Speed :

When a particle moves with different uniform speeds u1, u2, u3 etc. in different (finite) time intervals t1 , t2 , t3 etc. respectively, its average speed over the total time of journey is given as

$\displaystyle u_{av} = \frac{Total \; distance \; covered}{Total \; time \; elapsed}$

$\displaystyle u_{av} = \frac{S_1 + S_2 + S_3 + …}{t_1 + t_2 + t_3 + …..}$

$\displaystyle u_{av} = \frac{u_1 t_1 + u_2 t_2 + u_3 t_3 + —-}{t_1 + t_2 + t_3 + …..}$

#### Average Velocity :

When a particle moves with different velocities etc. in different time intervals t1 ,  t2 ,  t3 etc. respectively, its average velocity over the total time of motion can be given as

$\displaystyle u_{av} = \frac{Net \, displacement \, vector}{Total \, time }$

$\displaystyle \vec{u_{av}} = \frac{\vec{S_1} + \vec{S_2} + \vec{S_3} + ….}{t_1 + t_2 + t_3 + ….}$

$\displaystyle u_{av} = \frac{\vec{v_1} t_1 + \vec{v_2} t_2 + \vec{v_3} t_3 + ….}{t_1 + t_2 + t_3 + ….}$

Solved Example : A particle moves with a velocity v(t) = (1/2)kt2 along a straight line. Find the average speed of the particle in a time T.

Solution : $\displaystyle u_{av} = \frac{1}{T}\int_{0}^{T} v dt$

$\displaystyle u_{av} = \frac{1}{T}\int_{0}^{T} \frac{1}{2}k t^2 dt$

$\displaystyle = \frac{1}{6}k T^2$

Solved Example : Find the average speed of a particle whose velocity is given by v = vo sin ωt

Solution: Time Average

$\displaystyle v_{av} = \frac{1}{T}\int_{0}^{T} v dt$

$\displaystyle a_{av} = \frac{1}{T}\int_{0}^{T} a dt$

Here, v and a are the functions of  t .