Motion: the direction of velocity and acceleration

[Chuquis]

I am looking at motion and learnt that the velocity can negative while the acceleration positive,

 

so i wondered whether the velocity and acceleration could be in different direction or did the have to be both in the same Direction?

 

And I also have another question:

 

What does it mean when the acceleration is negative? I didn't understand the textbook.

[timo]

Velocity and acceleration can indeed be in different directions. If they are in opposite directions, the process is usually called "deceleration" or "breaking". The meaning of an acceleration (or a velocity) to be negative is the following: Acceleration (and velocity) have a direction. In many cases, the direction, that in principle can point anywhere, is simplified to "forwards" and "backwards", where the exact meaning of the two terms usually is clear from the context (think of a car, for example). Usually, positive values of acceleration/velocity mean that it is into the "forward" direction. Negative values mean that the direction is into the "backwards" direction. In textbooks, when you learn about velocity and acceleration you usually also have a position x. There, "forwards" usually means "towards increasing x", whereas "backwards" means "towards smaller x".

[Iggy]

I am looking at motion and learnt that the velocity can negative while the acceleration positive,

 

so i wondered whether the velocity and acceleration could be in different direction or did the have to be both in the same Direction?

 

Velocity is change in position. If something moves from zero, then one, two, and higher then the velocity is positive. If it moves the other way (3, 2, 1, then zero) the velocity is negative.

 

Acceleration is change in velocity. If it gets faster or slower over time then there is acceleration. If velocity gets larger (or more positive) over time then acceleration is positive. If the velocity gets smaller, or more negative, over time then acceleration is negative.

 

If it is speeding up then the acceleration vector is in the same direction as its motion. Slowing down means the acceleration is in the opposite direction. So you could imagine something moving very fast to the left (over time its position changes from larger to smaller), and it slows down over time. Velocity would be negative and acceleration positive.

 

And I also have another question:

 

What does it mean when the acceleration is negative? I didn't understand the textbook.

 

Negative acceleration just means that velocity is getting smaller over time. If something moves 4 meters to the left the first second then 2 meters in the second, and 1 in the third then it's acceleration was in the negative direction.

Edited January 31, 2013 by Iggy

[Crimson Sunbird]

As an illustration, let us consider a particle moving along the x-axis between [latex](-1,0)[/latex] and [latex](1,0)[/latex] in simple harmonic motion, its x-coordinate being given by the equation

[latex]x=\sin\frac{\pi t}2[/latex]

Its velocity and its acceleration are then given by

[latex]v=\frac{\pi}2\cos\frac{\pi t}2[/latex]

 

[latex]a=-\frac{\pi^2}4\sin\frac{\pi t}2[/latex]

For this motion, the velocity of the particle is positive when it is moving towards [latex](1,0)[/latex] and negative when it is moving towards [latex](-1,0)[/latex]. Its acceleration is positive when the particle is to the right of the origin ([latex]x>0[/latex]) and negative when it is to the left of the origin ([latex]x<0[/latex]). A summary of the motion is given below.

[latex]\begin{array}{c|c|c|c} \text{time} & \text{position} & \text{velocity} & \text{acceleration} \\ \hline 0\leqslant t\leqslant1 & 0\leqslant x\leqslant1 & \mathrm{+ve} & \mathrm{-ve} \\ \hline 1\leqslant t\leqslant2 & 0\leqslant x\leqslant1 & \mathrm{-ve} & \mathrm{-ve} \\ \hline 2\leqslant t\leqslant3 & -1\leqslant x\leqslant0 & \mathrm{-ve} & \mathrm{+ve} \\ \hline 3\leqslant t\leqslant4 & -1\leqslant x\leqslant0 & \mathrm{+ve} & \mathrm{+ve} \\ \end{array}[/latex]

Thus you can see that velocity and acceleration are opposite in sign except for [latex]1\leqslant t\leqslant2[/latex] and [latex]3\leqslant t\leqslant4[/latex]. At these time intervals, the particle is increasing in speed whereas it is slowing down in the other two time intervals. The acceleration is negative during the first half of the motion: the particle is experiencing a force directed to the left (whereas force on it is to the right when the acceleration is positive).