i have doubts in questions which include a moving elevator and a ball or an object being dropped inside it with some given speed/acceleration.

Dear Kaineet

you need not to worry.
you must know the concept of inertial and noninertial frames.non inertial frames have a special force called Pseudo Force acting on the object in a system opposite to the acceleration of the system.
when the elevator goes up by an acc=a and a ball is dropped at the same time then net acc. of the ball in the frame of elevator which is noniertial is =g+a
and vice versa.
it is also worth mentioning that only accelerating frames are noninertial.the non accelerating or frames moving with constant velocity do not account for any pseudo force.so the object or ball moves down by an acc=g only.

Regards.

Use the concept ofrelative motion and post a specific problem for us to help.

there are two cases.one,lift is not accelerating and two,it is accelerating

CASE 1:NO ACCELERATION

here,take acceleration of the ball as g or -g according to going up or down.

CASE 2:ACCELERATION

USE PSEUDO FORCE , THAT IS WHEN LIFT GOES UP ,TAKE g+a AND WHEN IT COMES DOWN TAKE g-a.

 concepts of relative velocity will help you out.