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

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Discussion Overview

The discussion revolves around the labeling of acceleration vectors on a velocity-time graph, particularly when an object is moving in the negative direction. Participants explore how to correctly denote the direction of acceleration when an object is speeding up or slowing down, considering the implications of positive and negative values in different coordinate systems.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes that if an object is speeding up in the negative direction (West), the calculated acceleration will be negative, raising the question of whether to label it [West] or [East].
  • Another participant clarifies that 'negative acceleration' refers to the component of acceleration pointing opposite to the positive direction of an axis, emphasizing that vectors should be understood in terms of direction and magnitude rather than signs.
  • A participant calculates an acceleration of 1 m/s² and questions if it should be labeled [West] when the object is slowing down in the West direction.
  • One participant asserts that the slope of the line indicates the component of acceleration in the Easterly direction, not the direction of the velocity vector.
  • There is a confirmation of an acceleration value of -5 m/s², with a discussion about whether to label it as [East] or to include the negative sign.
  • Another participant argues that using a negative sign is unnecessary when indicating direction, suggesting a consistent convention should be adopted.
  • Conflicting views arise regarding the interpretation of negative values, with one participant stating that "-5 units East" could be seen as "+5 units West," while others maintain that the direction should be labeled based on the vector's orientation.
  • Participants clarify that the acceleration during a specific time period is pointing West, leading to a consensus on labeling it as such.

Areas of Agreement / Disagreement

Participants express differing views on how to label acceleration vectors, particularly regarding the use of positive and negative signs versus directional labels. While some agree on the direction of acceleration in specific instances, there remains no consensus on the broader principles of labeling.

Contextual Notes

The discussion includes various interpretations of vector components and the implications of coordinate systems, with some assumptions about the definitions of positive and negative directions remaining unresolved.

cvc121
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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...
 
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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.
 
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]?
 
Last edited:
Here is the graph.
 

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cvc121 said:
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.
 
Ok, thanks. Just to confirm, is the acceleration between 7-8 seconds -5m/s2 [East]?
 
cvc121 said:
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.
 
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)
 
  • #10
nasu said:
The acceleration is pointing East.
In the time period to which the poster was referring (7-8 seconds) the acceleration is pointing West.
 
  • #11
andrewkirk said:
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.
 

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