Acceleration Question -- How to label the vectors of motion....

[cvc121]

On a velocity-time graph, if an object is speeding up in the negative direction (let's say West), the acceleration that is calculated will be negative. To show the direction, would we put [West] or [East]? Also, if the object is slowing down in the negative direction (let's say West again), would we put [West] or [East] after calculating a positive value for acceleration. A little bit confused about this...

 

[andrewkirk]

Strictly speaking, when we say that 'the acceleration is negative' we actually mean 'the component of acceleration in the direction I just recently mentioned points in the opposite direction to that direction'. That direction that has just been mentioned is usually the positive direction of an axis of a particular coordinate system. In your example, it's the positive direction of the x axis.

Vectors do not have positive or negative signs. They are better thought of as a specified direction and magnitude. Signs only arise when we look at the component of a vector in a given direction. When we represent a vector as a triple of numbers, the numbers can be negative, but the numbers are not the vector. They are just a representation of it in a particular coordinate basis. If you keep this in mind you can avoid confusion.

It is common in basic mechanics for people to talk about 'negative acceleration' but that is just a shorthand for saying the projection of the acceleration vector on the vector pointing along the positive directional axis points in the opposite direction to that axis. Or in symbols

$$\vec{a}\cdot \vec{e}_x<0$$

where the dot represents the vector dot product (aka inner product or scalar product) and $\vec{e}_x$ is the unit vector pointing in the positive direction of the x axis.

 

[cvc121]

Taking the time interval A (8-10s) for example, calculating the slope of the line gives us an acceleration of 1m/s². The positive direction in this case is East. Since the object is slowing down in the West direction, would I be correct in saying that a= 1m/s² [West]? If it was speeding up in the West direction, a= -1m/s² [West]?

 

[cvc121]

Here is the graph.

 

[andrewkirk]

would I be correct in saying that a= 1m/s2 [West]?

No. The slope of the line in that region is the component of the acceleration in an Easterly direction. So acceleration is $1ms^{-2}$ Easterly. It is the velocity vector that points West in that region of the graph, not the accel vector.

 

[cvc121]

Ok, thanks. Just to confirm, is the acceleration between 7-8 seconds -5m/s² [East]?

 

[andrewkirk]

Yes

 

[nasu]

Ok, thanks. Just to confirm, is the acceleration between 7-8 seconds -5m/s² [East]?

The acceleration is pointing East. There is no point to put a minus there. You are using E and W to indicate direction and not plus and minus. You have to pick one convention and stay with it.

 

[jtbell]

If I saw something like "-5 units East", I'd interpret it as equivalent to "+5 units West". One way such a thing could arise is in subtracting one vector from another:

(5 units East) - (10 units East)
= (-5 units East)
= (+5 units West)

 

[andrewkirk]

The acceleration is pointing East.

In the time period to which the poster was referring (7-8 seconds) the acceleration is pointing West.

 

[nasu]

In the time period to which the poster was referring (7-8 seconds) the acceleration is pointing West.

You are right. Then just put the direction as West.