- [Voiceover] The spring is now compressed twice as much,
to delta x equals 2D.
A student is asked to predict whether the final position
of the block will be twice as far at x equals 6D.
The student reasons that since the spring will be compressed
twice as much as before, the block will have more energy
when it leaves the spring, so it will slide farther
along the track before stopping at position x equals 6D.
So, let's just think about what the student is saying
or what's being proposed here.
So, in the first version, the first scenario, we compressed
the block, we compressed the spring by D.
And then, the spring accelerates the block.
And then, right when we get back to x equals zero,
all of that potential energy has been turned into
kinetic energy.
And then, the friction is acting
against the motion of the block,
so you can view it as it's providing negative work.
The direction of the force is opposite to the change in x.
And the negative work eventually causes the block to stop.
And all of that kinetic energy has now turned into heat.
And so, the block goes 3D.
Now, this new scenario, we could call that scenario two,
we are going to compress the spring twice as far.
So, now we're gonna compress the spring twice as far.
So, we're gonna compress it by 2D.
So, this is x equals negative 2D here.
And what's being said, or what's being proposed,
by the student is alright, if we compress it twice as far,
all of this potential energy is then going to be,
we're definitely going to have more potential energy here
because it takes more work to compress the spring that far.
And then, all of that more potential energy is gonna
be converted to more kinetic energy once we get back
to x equals zero.
And so, not only will it go further, but they're saying
it'll go exactly twice as far.
So, we are going to go, instead of going to 3D, we are
now going to go to 6D.
Now, let's read.
Let's see what the questions are here.
And actually, I'm gonna put a question mark here since
I'm not sure if that is exactly right.
So, part (b) i., let me do this.
So, we're in part (b) i.
It says which aspects of the student's reasoning, if any,
are correct.
Explain how you arrived at your answer.
And then, part two says which aspects of the student's
reasoning, if any, are incorrect.
Explain how you arrived at your answer.
So, the student is correct that two times,
so compressing more,
compressing spring more,
spring more, will result
in more energy when the block leaves the spring,
result in more energy
when block leaves the spring,
block leaves spring,
which will result in the block
going further, which will result,
or the block going farther I should say,
which will result in longer stopping distance,
which will result in longer stopping
stopping distance.
Now, part two.
Part two, here.
Which aspect of the student's reasoning, if any,
are incorrect.
Explain how you arrive at your answer.
Well, two times I could say, let me say compressing,
compressing twice as much,
twice as much, does not result
in exactly twice the stopping distance,
does not result in
twice the stopping distance,
the stopping distance.
And we can explain more if we like.
We know that potential energy is equal to 1/2 times
the spring constant times how much we compress, squared.
So, we could say that energy,
energy grows
with the square,
of compression of how much
we compress it.
of how much
we compress.
So, two times the compression.
I'm gonna say two times.
I'll write it out, two times compression
will result in four times
the energy.
Energy.
And this will result in four times the stopping distance,
four times stopping distance,
four times stopping,
stopping, distance.
I think that it does a decent job of explaining where
the student is correct, where their reasoning is correct,
and where it is incorrect.