In A2/A-Level Mathematics, we will discuss non-uniform acceleration.

Theory

In A2/A-Level Mathematics, Consider a particle P moving in a straight line from a starting point O.

The displacement from O is *x *at time *t *. The initial conditions are: *t *≥ 0 when *x*=0. if *v *is the velocity of P at time *t*, then

**v = dx/dt**

The acceleration '*a*' of particle P is defined as:

**a = dv/dt =d ^{2}x/dt^{2}**

or alternately,

**a = dv/dt **

**= (dv/dt)(dx/dt)**

BUT **v = dx/dt**

** a = v(dv/dx)**

In A2/A-Level Mathematics, Problems on this topic are solved by analysing the information given to form a differential equation. This is then integrated, usually between limits.

__Example #1__

A particle moves in a straight line such that its acceleration '*a*' at time '*t*' is given by:

a = 4t-7

If the initial speed of the particle is 5 ms^{-1}, at what values of '*t*' is the particle stationary?

__Example #2__

A particle moves from a point O in a straight line with initial velocity 4 ms^{-1}.

if *v *is the velocity at any instant, the acceleration *a *of the particle is given by:

a = 3/v

The particle passes through a point X with velocity 8 ms^{-1}.

i) how long does the particle take to reach point X?

ii) what is the distance OX?(1 d.p.)

i)

ii)

That's the end of the topic in A2/A-Level Mathematics.