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Im self studying physics and came to this question in my textbook:

A passenger in a moving bus with no windows notices that a ball that had been at rest on the aisle suddenly start to move towards the rear of the bus. Think of two different possible explanations and devise a way to decide which is correct.

Now the two possible scenarios are clearly (1) the bus is speeding up and (2) The bus starts traveling uphill. But I do not know what experiment should be devised to see which it is. I feel like it has something to do with inertial frames. i.e. the first option for the motion of the ball is because the inertial frame is accelerated whereas the second is because all of a sudden the force of gravity began acting on the ball in the horizontal direction (w.r.t. the surface of the bus).

But I am having a hard time thinking of an experiment that can deduce which is occurring. Obviously one "experiment" is look at the speedometer, but I have a feeling that is not what the point of the question is. I was thinking of maybe dropping the ball? And then observing its flight path? If the bus is accelerating then the ball will fall straight down whereas if the bus is traveling uphill then the ball will have a slanted path toward the ground. Would this work and/or is there a better approach?

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Given the two scenarios:

  1. The bus is moving at steady state speed on a flat ground and suddenly accelerates forward
  2. The bus is moving at a steady state speed on a flat ground and suddenly encounters an upward slope

You can note that the force diagram on the ball would be different given each situation.

If we are living in scenerio (1), you would expect that when hung from a scale that could rotate freely (like those you can find at a grocery store), the scale would tilt towards the back of the bus, and show a force higher than simply the mass of the ball itself. This is because the scale has to resist not only the acceleration due to gravity, but also the extra horizontal acceleration from putting your foot on the gas pedal (and you could use the pythagorean theorem to calculate this increase in force, due to the vectors being orthogonal).

However, if we are living in scenario (2), you would imagine that this same rotating scale would simply tilt towards the back of the bus, but read no increase in force from the ball. This is because we are moving at steady-state speed, and the coordinates of our system are merely rotated due to the hill.

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  • $\begingroup$ But what if we did not have a scale? My guess is the solution here would only require the ball and like our bodies. $\endgroup$
    – Chris Christopherson
    Commented Dec 8, 2023 at 22:16
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    $\begingroup$ "devise a way to decide which is correct" is pretty open-ended; id read it as saying you can have pretty much equipment you can dream of. Though I suppose if you were attentive enough, you might be able to feel the difference in weight just by holding the ball in your hand (but I'm guessing this would be pretty tricky to get consistent in reality) $\endgroup$
    – Matteo
    Commented Dec 9, 2023 at 0:18
  • $\begingroup$ You may be right. Either way clever idea/solution ;) $\endgroup$
    – Chris Christopherson
    Commented Dec 9, 2023 at 1:57
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Measure the acceleration with an accelerometer (or measure the effective g by measuring the period a pendulum), and the direction in which a pendulum at rest hangs from the ceiling. With those two measurements and some algebra (that relate those quantities to $a$ and $\theta$) you can recover both, the angle of the ramp is moving up, and the acceleration is moving up, if any.

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