Question about this Acceleration vs. Time graph

[someonehere]

Homework Statement

My books says that, when a body is moving with constant decreasing acceleration, the acceleration-time graph is a straight line. Then, it provides a graph (attached below).

Relevant Equations

According to the graph, acceleration is directly proportional to time and the constant of proportionality is negative. If it is directly proportional, why doesn't it pass through the origin?

 

[kuruman]

According to the graph, acceleration is a linear function of time. It is not directly proportional to time. Directly proportional means that if you multiply time by a factor $f$ the acceleration is multiplied by the same factor. This is not the case here. I don't know why they have $a=-kt$ on the graph.

 

[Gavran]

Acceleration=tgθ⋅Time+C
Time≠t
Acceleration≠a
Time=c₁⋅a+c₂⋅t+c₃
Acceleration=c₄⋅a+c₅⋅t+c₆
tgθ=(c₅-k⋅c₄)/(c₂-k⋅c₁)
C=c₆-c₃⋅(c₅-k⋅c₄)/(c₂-k⋅c₁)

 

[kuruman]

Acceleration=tgθ⋅Time+C
Time≠t
Acceleration≠a
Time=c₁⋅a+c₂⋅t+c₃
Acceleration=c₄⋅a+c₅⋅t+c₆
tgθ=(c₅-k⋅c₄)/(c₂-k⋅c₁)
C=c₆-c₃⋅(c₅-k⋅c₄)/(c₂-k⋅c₁)

That's a series of equations with a whole lot of symbols that you have not defined or explained. It is impossible to evaluate what you are trying to say or even understand it. I suspect that you are trying to write a straight line equation for the graph shown in the picture. If that is the case, then you need 4 and only 4 symbols because a straight line has the form $$y=mx+b$$where
$y = ~$ the dependent variable read on the vertical axis. Here it is labeled "Acceleration."
$x = ~$ the independent variable read on the horizontal axis. Here it is labeled "Time."
$m = ~$ the slope. Here you can read from the graph as $\tan\!\theta$.
$b = ~$ the $y$-intercept (value of $y$ at $x=0$). Here, the graph does not define it in any way.

So if your goal is to write a straight line equation based on what is shown on the graph, you should have $$\text{Acceleration}=\tan\!\theta*\text{Time}+b$$and leave it at that. If that is not your goal, then please explain what it is. As mentioned earlier, I don't know what $a=-kt$ on the graph is supposed to indicate.

 

[Gavran]

That's a series of equations with a whole lot of symbols that you have not defined or explained. It is impossible to evaluate what you are trying to say or even understand it. I suspect that you are trying to write a straight line equation for the graph shown in the picture. If that is the case, then you need 4 and only 4 symbols because a straight line has the form $$y=mx+b$$where
$y = ~$ the dependent variable read on the vertical axis. Here it is labeled "Acceleration."
$x = ~$ the independent variable read on the horizontal axis. Here it is labeled "Time."
$m = ~$ the slope. Here you can read from the graph as $\tan\!\theta$.
$b = ~$ the $y$-intercept (value of $y$ at $x=0$). Here, the graph does not define it in any way.

So if your goal is to write a straight line equation based on what is shown on the graph, you should have $$\text{Acceleration}=\tan\!\theta*\text{Time}+b$$and leave it at that. If that is not your goal, then please explain what it is. As mentioned earlier, I don't know what $a=-kt$ on the graph is supposed to indicate.

The question is why the graph a=-k*t does not pass through the origin on the picture. Because the function a=-k*t is not represented in the coordinate system with axes t and a, but represented in the coordinate system with axes Time and Acceleration. Axes t and a are transformed into axes Time and Acceleration.
It is obvious from the picture, where the graph does not pass through the origin, that a coordinate transformation which is called the translation (Time=t+c3, Acceleration=a+c6) exists in this case.
I included linear transformation (Time=c2*t+c1*a, Acceleration=c5*t+c4*a) just in case, although there is not enough information in the problem statement for making conclusion if this kind of coordinate transformation exists or does not exist.

 

[kuruman]

The question is why the graph a=-k*t does not pass through the origin on the picture.

And the answer to that is simple: Because the equation $a=-kt$ is not the mathematical equation for the line shown in the picture. Your book provided the wrong equation for the line shown in the picture. To match the equation, it should have shown the straight line representing the acceleration crossing the time axis at the origin O.