Acceleration and motion can be in different direction? I'm not getting what acceleration concept is and how it relates to motion and how motion and acceleration can be in different direction? And what's behind the concept of negative and positive acceleration?
 A: Let's say that we move along the straight line. Acceleration shows how fast velocity changes, it doesn't matter how fast you move:


*

*If velocity increases, acceleration is positive

*If velocity doesn't change, acceleration is 0

*If velocity decreases (slows down), acceleration is negative


So when you're in the car and you step on brakes, you keep moving forward for some time, but acceleration is negative (points backwards) - it opposes the forward motion. 
If you want an image in you head - instead of brakes think of Hulk stopping the car - he pushes it into opposite direction, the car keeps moving but slows down.
A: 
How motion and acceleration can be in different direction?

This isn't surprising. Hitting the brakes on your car is not the same as putting it in reverse. 
A: Think of a satellite in orbit. At any point in time it is moving "horizontally" (tangential to the earth). However, its acceleration is always directly towards the centre of the earth, in other words, at $90^{\circ}$ to its direction of motion.
"Motion" is how the object is currently moving. Acceleration can be in any direction; it depends on the direction of the force. For a satellite, the only force is the earth's gravity, the direction of which is towards the centre of the earth.
A: *

*The plus/minus sign of position, velocity, and acceleration are applicable only to one-dimensional motion.


*Position vector's sign refers to the object's location relative to the reference point;
velocity's sign refers to the object's direction of movement;
acceleration's sign refers to the direction of the net force $(F=ma)$ causing the object to accelerate.


*҂Incidentally, displacement (change in position, i.e., final position vector $-$ initial position vector) has the same direction as average velocity.
(Position might be denoted by $x$ or $r,$ while displacement by $s.)$


*To see how velocity (motion) and acceleration can be in opposite directions, consider a ball travelling upwards while its acceleration (due to gravity and air resistance) is downward.


*To dispel a common misconception: an object having a negative acceleration is not necessarily undergoing retardation! Retardation is the rate of decrease of speed, not velocity.
To wit: define the positive direction as upward, and consider a ball in free fall. Then its acceleration (due to gravity) is negative but it is not retarding (it is speeding up).


*The term ‘deceleration’ is best avoided due to potential ambiguity; however, it is often defined as retardation instead of as negative acceleration, which is not a meaningful concept outside of one-dimensional motion.
As such, in the previous example (#5), the ball's acceleration and deceleration are both negative; in the first example (#4) where the ball is slowing down, if we define the positive direction as downward, then the ball's acceleration and deceleration are both positive.
Here, while acceleration is a vector whose sign refers to its spatial direction, retardation/deceleration is merely a signed scalar whose sign refers to the direction of speed change (positive deceleration corresponding to decreasing speed).
҂ (i) position $$= \int v\:\mathrm dt;$$
(ii) displacement = average velocity $\times$ time elapsed $$= 
\int^{t_2}_{t_1} v\:\mathrm dt;$$
(iIi) distance travelled = average speed $\times$ time elapsed $$= \int^{t_2}_{t_1} |v|\:\mathrm dt.$$
A: Try to remember the following things in short which will help you to develop this idea of your own.
(A) Acceleration is in the direction of motion, when you are increasing the speed of the car moving along a straight line.
(B) Acceleration is against the direction of motion, when brakes are applied to the car in motion.
(C )Acceleration is uniform, when an object is under free fall.
(D) Acceleration is non-uniform when a vehicle is driving through a busy city road.
Hope this will help.
