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?

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.

• You don't seem to know that acceleration is a vector... – Matt Jun 30 at 10:16
• The complexity of an answer should correspond to the level of a question. Otherwise only people who know the topic anyway would understand the explanation. – Stanislav Bashkyrtsev Jun 30 at 11:01
• Am I dreaming or did you edit OP's question to add "And what's behind the concept of negative and positive acceleration"??? – Matt Jun 30 at 18:58
• I added words "And what's behind" instead of just "what" to make it grammatically correct. Why? – Stanislav Bashkyrtsev Jun 30 at 19:02

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.

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.

$$\vec{F} = m\,\vec{a}$$

Here Symbols have usual meaning. Now consider the left hand side of equation. It gives Force acting on particle. It can be Gravitational or Electromagnetic Forces (or Nuclear Forces too but we keep ourselves restricted to the former two.

Now right hand side gives the acceleration of particle. Acceleration is defined as:

$${a }= \frac{d^2 x}{dt^2}$$

Now let's take mass to be positive.

Then the first equation says that Force acting on body is in the direction of acceleration of the body on which force is applied. Further notice that mass is scalar quantity and the Force and acceleration are vector quantities. Then Force formula simply relates the force in each direction to acceleration of body in that direction.

Now consider the spring force acting on body (for brevity consider S.H.M motion) Then Hooke's law applied to mass (on which spring force is applied) gives:

$$\vec{F} = -k\,\vec x$$

Now, the body is moving forward but force is acting backward on the object. That is object is retarding. Therefore we have:

$$-k\,\vec{x} = m\,\vec{a}$$

Which says force is in direction opposite to retardation or equivalentaly since retardation is negative of acceleration, force is in the direction of acceleration. If the particle is moving away from origin, then the above equation says that particle is pulled in the direction toward origin and hence its acceleration is also towards origin.That is the body is slowing down.

Hope this helps

• Which says force is in direction opposite to acceleration. Not at all. This can never be, since mass is positive. That is the body is slowing down. You're confusing displacement and velocity. You'd better to understand things yourself, before answering someone's questions. – Elio Fabri Jun 30 at 14:29
• @ElioFabri Edited and clarified with example – Abhi7731756 Jul 1 at 1:45