# Acceleration of a particle moving along a straight line

If a particle moves along a straight line, then can we say that the acceleration should be zero? and if not, then why?

If the equation of the motion is linear, then the velocity is constant and if velocity is constant, then acceleration is zero.

• Hi. Why if a particle moves in a straight line it's acceleration will be zero? May 12 '15 at 9:53
• I just gave my reasoning at the bottom of my question. Because acceleration is the derivative of velocity and constant acceleration gives zero velocity. I am surely wrong and that is why I am trying to see what is wrong with my logic. May 12 '15 at 9:57
• If motion is in one dimension (along a line or linear), that doesn't mean that the expression of that motion will be with a linear equation. The two meanings of linear are distinct here. May 12 '15 at 9:57
• Constant acceleration means constant change of the velocity in time, that is a changing with constant rhythm velocity. As a derivative of velocity, acceleration, even if is zero, doesn't mean you don't have velocity. In general, if the velocity is a function of time, the price of the variable t(time) that gives zero acceleration will give a max or low of velocity. If the velocity is a linear function of time, then zero acceleration gives constant velocity(and if the starting condition demand it, zero velocity-that means the particle had zero valocity). May 12 '15 at 10:02
• @user77791: No, it's the opposite. A constant velocity means zero acceleration. You have to integrate the acceleration to get the velocity.
– GRB
May 12 '15 at 10:27

You are using the word "linear" in two different ways.

When an object moves along a straight line we can say its motion is linear - but that does not mean its acceleration is zero. Just that the acceleration points along the same direction as the velocity (so no change in the direction of the motion).

The second meaning of "linear" is in the exponents of the mathematical terms for the equation of motion - either time or position, for example.

The following equation describes linear motion with acceleration:

$$\vec r(t)= (a\cdot t^2, 0)$$

This is uniform acceleration along the X axis. It is "linear" in the sense of moving along a line.

Now if position is a linear function of time (which is a much narrower reading of "linear motion"), then and only then can you say the velocity is constant and the acceleration is zero.

Velocity is a vector quantity having both magnitude and direction. Change in either one of these or both is acceleration. A body moving along a straight path would have constant direction but might have variable speed.

Consider a mass which is dropped from a height. This is done in a vacuum, or is otherwise shielded from crosswinds. Furthermore, the mass is located on a non-rotating body, has no lateral initial velocity, and is not noticeably affected by the gravitation of any other bodies.

Two things are clear: first, the mass moves in a straight line toward the center of the body; second, it accelerates along that straight line.

The phrase "a particle moves along a straight line" may or may not be used to describe motion described by a linear equation, and in the more general sense does not imply this meaning. To describe a path as being governed by a linear equation usually means something like (for 2D paths) y = ax + b. Note that this description of the path says nothing about the velocity of a point along the path - it might be constant, or it might not.

If your direction of motion is linear it does not necessary mean that the equation representing its position with respect to time is always linear.And if the velocity is constant it always means that acceleration is zero.