Meeting the criteria for a graph

[Stendhal]

Homework Statement

The cart is initially at rest. The overall goal will be to determine the cart’s acceleration. Plot the data in such a way that:
The position is the independent variable, and plotted on the horizontal axis.
The acceleration is inversely proportional to the slope.
The resulting plot is a straight line.. (Start with x = ½ a t^2 and use algebra )

There will be a table of the distance and time of the car.

Homework Equations

x = ½ a t^2

The Attempt at a Solution

My biggest confusion with this is what will be on the vertical axis. If the acceleration is inversely proportional to the slope, then wouldn't the units of acceleration just be flipped in order to match the slope? Basically, wouldn't the vertical axis be time squared? That doesn't make any sense to me, as time is generally not on the vertical axis, but it also cannot be in the horizontal axis as the criteria asks that the position is the horizontal axis.

I think in order to meet the 3rd point, I must make the original equation into a natural log plot, where:
ln(x) = ln(½) + ln(a) + ln(t^2). Would that be the correct way to go about this?

Any and all help is deeply appreciated!

 

[andrewkirk]

You haven't said what ground the cart is going over. Reading between the lines, it sounds as though it will be on a slope of constant gradient. If that's not correct, more info would be needed.
It is unusual in Introductory Physics to put time on the vertical axis, but not forbidden. In Special Relativity, graphs with time on the vertical axis are the norm.
Your suggestion of using log plots would work, but is more complex than it needs to be, as it involves transforming both the axes. A linear plot can be achieved with a simple transformation of only one axis.
If you put untransformed time on the vertical axis and position on the horizontal, and acceleration is constant, what transformation do you need to apply to the horizontal axis to make the line of the cart's position straight?

 

[Stendhal]

You haven't said what ground the cart is going over. Reading between the lines, it sounds as though it will be on a slope of constant gradient. If that's not correct, more info would be needed.
It is unusual in Introductory Physics to put time on the vertical axis, but not forbidden. In Special Relativity, graphs with time on the vertical axis are the norm.
Your suggestion of using log plots would work, but is more complex than it needs to be, as it involves transforming both the axes. A linear plot can be achieved with a simple transformation of only one axis.
If you put untransformed time on the vertical axis and position on the horizontal, and acceleration is constant, what transformation do you need to apply to the horizontal axis to make the line of the cart's position straight?

I am not sure if it applies, but in the sheet we were given, an earlier experiment was to find the acceleration of a cart that was on a 5 degree slope. My original question is a separate problem, so I don't think it applies. However, I see what you are trying to get at in asking that. Unfortunately, that information isn't given in this particular problem.

Can I assume that the untransformed vertical axis would be t^2 as I originally stated? If that is the case, then it would seem that the acceleration could be found by manipulating the above equation so that the transformation I would have to make for the horizontal position would only have to be to double it. Is that correct?