- [Instructor] Blocks A and B are pressed together
with a spring between them.
When the blocks are released from rest,
the spring pushes the blocks apart
so that the 75/100 of a kilogram block A
moves up the 30 degree ramp to the left
and the 0.25 kilogram block B
moves to the right at a velocity of three meters per second.
Assume friction is negligible.
Which of the following is the best estimate
of the maximum distance d that block A slides up the ramp?
So pause this video and see if you can figure that out.
All right now let's just first think about
what would happen intuitively.
So we have A and B, could imagine you're squeezing
them together so you have that energy stored
in that spring.
And then when you let go,
we know that B moves to the right
and you would intuitively think that A would move
to the left.
And it seems like the law of conservation of momentum
is going to be at play here.
Because if the system of A and B is initially at rest,
each of these blocks' velocities are zero,
doesn't matter what their masses are.
That means that the total momentum for the system
of A and B is going to be zero.
So even after we let go and B moves to the right
and A moves to the left,
the total momentum of this system
should still be zero.
And so maybe we can use that to figure out
what A's velocity is as it's sliding
to the left horizontally.
And then using that velocity,
well we could think about what's its kinetic energy.
And if we know its kinetic energy,
we could think about well,
as it goes up the ramp,
that kinetic energy's going to be turned
into potential energy.
And so how high up the ramp would it go?
And then from that and that angle,
we could figure out our distance.
So let's do that.
So let's start off with the law of conservation of momentum.
We know that after releasing,
A's momentum plus B's momentum
should add up to zero
because the sum of their momentums was zero
when they were at rest together,
when they were compressed.
So what we could say is the momentum for A
is going to be A's mass,
so let me write it right over here,
so 0.75 kilograms is A's mass
times its velocity.
Well we don't know what this is.
Let's call this velocity sub
or final velocity for A,
and that's going to be in meters per second,
plus B's mass, which is 0.25 kilograms,
times B's velocity.
Well that's going to be three meters per second,
times three meters per second.
This should be equal to zero.
And so now we just have to solve for that.
And so what do we get?
Let's see, if we subtract this part
from both sides,
we will get,
and this is just 0.75 kilogram meters per second,
so we'll get 0.75 kilograms
times the A's velocity after it gets released
is equal to negative 0.75
kilogram meters per second.
And so if you divide both sides by 0.75 kilograms,
on the left, you're just going to be left with
the velocity of A,
and on the right,
you would be left with negative one,
then the kilograms would go away,
negative one meters per second.
So A's velocity as it lets go from the spring
or as it loses contact with that spring
is going to be negative one meters per second.
Now what does the negative mean?
Well if to the right is positive three meters per second,
then to the left is going to be negative.
So it's going to be moving to the left
at one meter per second.
So let's use that information to think about
how far up the ramp it's going to go,
and we're assuming that friction is negligible
and nothing really interesting happens here,
although we know in the real world it might bump
a little bit, but we're just going to assume
that that's not going to be a major factor.
And so all of that kinetic energy
when it's moving to the left at one meter per second
is going to be converted to potential energy at some height.
So let's say that once all of the kinetic energy
turns into potential energy,
A is at an altitude of h.
So what's its potential energy going to be
at that point in time?
Well it's going to be the mass of A
times the gravitational field,
roughly 9.8 meters per second squared,
if we wanna be even rougher
say 10 meters per second squared,
times that height.
So that's going to be once all of our kinetic energy
has been turned into potential energy.
Well this should be equal to the amount
of kinetic energy we had at this point right over here.
Well what is that going to be?
Well that's going to be 1/2 m v squared.
So it's 1/2 times the mass of A
times the velocity of A squared.
So the velocity is negative one meters per second,
and then we're going to square it.
Well we could divide both sides by the mass of A.
Negative one squared is just going to be a one.
So this is just going to be a one eventually.
And so you're left with and then we can divide
both sides by g,
and we see that our height is equal to
1/2 and then we're going to have,
divided by g,
divided by g, and then this is going to be meters.
And so what is this going to roughly be?
Well h is going to be approximately,
let's say that g is approximately 10 meters per second,
because they talk about estimating here.
So this would be one over two times 10,
and then we're gonna get meters.
So this is gonna be 1/20 of a meter.
So h is approximately 0.05 meters.
So you might be tempted to say oh look
there's an answer there,
but remember they're not asking us this h.
They're asking us this d.
And so here we just you have to use
a little bit of trigonometry.
Where we say okay what trig ratio deals
with the opposite of an angle and the hypotenuse?
This is a right triangle right over here.
Well that would be sine.
So sine of 30 degrees,
sine of 30 degrees,
would be equal to the opposite side which is h over d.
We know that h is 0.05.
Sine of 30 degrees,
if you don't have a calculator this is a good one to know,
is 1/2, so you have
1/2 is equal to 0.05 divided by d.
We can multiply both sides by d,
and then you'll have d over two
is equal to 0.05
or multiply both sides by two and you get d
is equal to 0.10
or I could say roughly equal to
since we are estimating.
We had an estimate for g there.
And then this is choice D right over there.