physics problem (hookes) thinking too deeply?

[blackmi4]

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"In many harbors, old automobile tires are hung along the sides of wooden docks to cushion them from the impact of docking boats. the tires deform in accordance with Hooke's law. as a boat is brought to a stop by gently colliding with the tires, the rate of deceleration of the boat is:"

The answer is "increases until the boat stops"

because

"The force changes with the displacement of the tires. The greater the displacement, the greater the force, the greater the magnitude of acceleration as per the formula F=-kdx=ma"



**My issue is this. The boat would stop after some deformation/displacement of the tires and the tires would push the boat back. That is, after that maximum deformation/displacement is reached the tire would start regaining its original shape and dx would be decreasing. Therefore, you would have decreasing deceleration.

So, wouldn't the better answer be "increases and then decreases before the boat stops"?

Maybe it is thinking too deeply but in physics I feel like we always speak of net displacement. The boat hasn't finished moving when the tires are experiencing maximum deformation/displacement just as a ball, thrown in the air, hasn't finished moving once it has reached its maximum height.

In terms of physics in general and the mcat, what do you think? Am I thinking too deeply here?

Thanks in advance.

 

[Schenker]

That is, after that maximum deformation/displacement is reached the tire would start regaining its original shape and dx would be decreasing.

If dx is decreasing, then by definition the boat is moving away from the dock, meaning dx can only decrease after the boat has come to a stop. In other words, maximum deformation/displacement coincides with when the boat stops.

 

[blackmi4]

If dx is decreasing, then by definition the boat is moving away from the dock, meaning dx can only decrease after the boat has come to a stop. In other words, maximum deformation/displacement coincides with when the boat stops.

Thank you for the reply.

However, the boat stops twice.

First the boat stops FIRST when the tire has reached maximum deformation.

Then the boat starts moving away from the dock as you say. So the boat hasn't really stopped yet.

The tire pushes back on the boat as it regains its original shape (with dx decreasing).

The boat stops for the SECOND time at about the same time as the tire regains its original shape.

I think of it as the boat moving 12 meters west, then 0.5 meters east. It stopped at 12 meters, was pushed back, and stops again at 11.5 meters (when the tire has regained its shape).

So I am looking for someone to tell me if you see the same thing and agree. I feel like this may be overthinking a bit, but for someone who hasn't taken the mCAT, I wouldn't know.

Thanks again!

 

[Schenker]

I think you're overthinking it. Yes, in an ideal system with a spring that obeys Hooke's Law and with the only frictional forces on the mass being drag, the boat would do as you say. However, the question does not ask what happens to the boat after it stops [for the first time].

Since the question says "as a boat is brought to a stop" without specifying which stop, it's safer to assume that the question refers to the first stop. After the boat stops (for the first time), the question is done.

The smoking gun that the question refers the first stop is the word "deceleration."

as a boat is brought to a stop by gently colliding with the tires, the rate of deceleration of the boat is

Here, usage of "deceleration" implies that the correct answer will only consider the motion of the boat as it slows down; after the first stop, the boat will speed up, and usage of "deceleration" would not be appropriate. If the question were asking you about the sequence of events leading up to the second stop, the question would not have specified "deceleration" and instead would have said "acceleration" or simply "dv/dt."

A question like this on the MCAT might have included a diagram to clarify. However, as you've probably seen on practice tests, and as you will definitely see on the actual test, there will be a handful of problems such as these that lend themselves to misinterpretation and overcomplication. That's why you've probably heard that verbal reasoning carries over to the science sections.

 

[blackmi4]

I think you're overthinking it. Yes, in an ideal system with a spring that obeys Hooke's Law and with the only frictional forces on the mass being drag, the boat would do as you say. However, the question does not ask what happens to the boat after it stops [for the first time].

Since the question says "as a boat is brought to a stop" without specifying which stop, it's safer to assume that the question refers to the first stop. After the boat stops (for the first time), the question is done.

The smoking gun that the question refers the first stop is the word "deceleration."



Here, usage of "deceleration" implies that the correct answer will only consider the motion of the boat as it slows down; after the first stop, the boat will speed up, and usage of "deceleration" would not be appropriate. If the question were asking you about the sequence of events leading up to the second stop, the question would not have specified "deceleration" and instead would have said "acceleration" or simply "dv/dt."

A question like this on the MCAT might have included a diagram to clarify. However, as you've probably seen on practice tests, and as you will definitely see on the actual test, there will be a handful of problems such as these that lend themselves to misinterpretation and overcomplication. That's why you've probably heard that verbal reasoning carries over to the science sections.

Thank you, this helped me out a lot!