# Is acceleration causing displacement or velocity?

When an object is not moving and we affected it by some force does the given velocity cause the motion or is the motion caused by the acceleration?

Let us say that we have an object that is not moving so to make it move we apply some force which means we have just applied an initial velocity that is clear to me the object was not moving and after applying the force it starts move which means it gains some acceleration now is it the acceleration that causing that displacement or is it the velocity or the applied force?

• It causes a change in velocity. Velocity causes a displacement. – Ryan Unger Oct 19 '15 at 14:36

My answer will assume the reader knows some calculus, but I will try to explain the concepts in an intuitive manner as well.

Most of the question is about what we call "kinematics", that is the study of the motion of objects under the assumption the velocity/acceleration is known. Kinematics of point masses (in one dimension) boils down to these two equations: $$v = \dot x$$ $$a = \dot v.$$ One can also write: $$a = \ddot x.$$ The dot above a variable refers to the time derivative. The time derivative is the rate of change of the quantity. In other words, in a small step of time the change in velocity is given by: $$\Delta v = a \Delta t$$ and the change in position is given by: $$\Delta x = v \Delta t.$$ So if there is a change in position, there must be a non-zero velocity. (To say, velocity causes the change in position is a bit dangerous from a philosophical point of view, rather the velocity describes that there is a change of position.)

As an example, the following situations are possible:

• The acceleration is non-zero, but the velocity is zero.
• The velocity is non-zero but the acceleration is zero (this is what we call uniform motion).

How to get the position, given the velocity in dependence of time? That is done by the integral (in a sense the inverse of the differentiation): $$x(t) = x(0) + \int_0^t dt'\, v(t').$$ The same equation holds for velocity and acceleration: $$v(t) = v(0) + \int_0^t dt' \,a(t').$$

Now, we will go on to the part about forces: Forces are connected to acceleration by the well known formula: $$F = ma.$$ So substituting this in the equations above we get: $$F = m\dot v = m \ddot x.$$ So force is the "cause" for acceleration, accelration is the rate of change of the velocity and if a particle has a velocity it will move.

Note, that is usually considerably harder to solve the equation $m\ddot x = F$ than to calculate the integrals for the position and velocity above. The reason is, that $F$ often depends of the position. Such an equation (where a quantity and derivatives of a quantity are related) is called differential equation.

In conclusion, acceleration is directly related to the forces acting on an object, if the forces stop to act, there is no more acceleration, but the object will continue to have its velocity (as per Newton's law of inertia) and thus continue to move. Also, applying a finite force for an infinitesimal amount of time will not change the velocity (as $a \Delta$ will be zero).

To add to Matt S's answer, it helps to understand that velocity $v$ is the rate of change of displacement $x$, or mathematically:

$v=\lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}=\frac{dx}{dt}$.

Similarly, the acceleration $a$ is the rate of change of velocity $v$, or mathematically:

$a=\lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t}=\frac{dv}{dt}$.

And by extension: $a=\frac{d^2x}{dt^2}$.

Sort of all of them.

The applied force gives an acceleration corresponding to F=ma, acceleration is by definition the change in velocity per unit time, and velocity is the change in displacement per unit time.

If you want to think about it another way, acceleration is the change in displacement per unit time per unit time.

If we apply force we give acceleration. The body was at rest, so if we apply force it accelerates and get some velocity, if we remove that force, it will keep its motion steadily without changing the velocity and if a body has some velocity then it should change it's position i.e. displacement .So all the force,acceleration, velocity and displacement are cause and result of each other.

I would just like to add a little disambiguation to the previous answers in response to some of the language used in the original post.

Position, velocity, and acceleration are quantitative descriptors of the motion of the object. We don't say that, for instance, "acceleration causes changes in motion" because acceleration rather describes the change in motion of the object. I also would not say that the velocity "causes changes in position", i.e. leads to a displacement, because velocity is rather the quantitative description of how the position is changing with time. (See Sebastian Riese's answer.)

It is instead the net force applied to the object that causes an acceleration, i.e. it is the force that leads to the change in velocity of the object. This is the very essence of Newton's Second Law.

Finally, in a very real sense, there is no cause for the change in position of the object, because it is the natural state of objects to continue moving in the direction that they were already moving---and with the same speed---unless acted upon by a net force. This is the very essence of Newton's First Law.