Work - Energy Theorem problem?

[Samei]

Homework Statement

Problem: [/B]A curved plate of mass M is placed on the horizontal, frictionless plane as shown. Small puck of mass m is placed at the end of the plate and given an initial horizontal velocity v. How high the puck will shoot up from the plate? Regard radius of curvature to be much smaller than the jump height and consider that the puck leaves the plate vertically.

Homework Equations

Conservation of Energy: KE1 + PE1 - Wf = KE2 + PE2 and Wf = deltaTME = (delta K + delta P)
Also, possibly Angular Speed: w = v/r

The Attempt at a Solution

[/B]
From the first equation, KE1 + PE1 - Wf = KE2 + PE2
So, ((mv1^2)/2) + (mgh)) - Wf = ((mv2^2)/2) + (mgh))
Now, at the first point, h = 0. Thus, PE1 = 0. Also, since delta K = ((mv2^2)/2) - ((mv1^2)/2) and delta P = mgh - 0:
This extends to ((mv1^2)/2) - (((mv2^2)/2) - (mv1^2)/2) + mgh))) = ((mv2^2)/2) + (mgh))
Simplifying all this makes: (mv1^2)/2) - (mv2^2)/2) + (mv1^2)/2) - mgh = ((mv2^2)/2) + (mgh)) or
2(mv1^2)/2 = 2(mv2^2)/2) + 2(mgh))
Removing 2m, (v1^2)/2 = (v2^2)/2 + 2gh

I need another equation. This is when I turn to the angular speed (?), at least possibly. Momentum is not conserved since friction is present, so I am not sure what equation to make. Any suggestions on my next steps, or if I am on the right track at least? Any help is appreciated!

Note: Do I have to apply Calculus techniques here? Is that why I'm missing? I'm a little rusty on that area.

 

[haruspex]

It isn't stated, but it seems to me you have to assume there's no friction anywhere, or it becomes insoluble.
You do need to consider the motion and mass of the plate.

 

[Samei]

It isn't stated, but it seems to me you have to assume there's no friction anywhere, or it becomes insoluble.
You do need to consider the motion and mass of the plate.

I know this sounds strange, but can I assume there is no friction or can I just denote it as u? From the picture, it seems as if there is some rugged surface involved, but I am with you. I've been racking my brain in coming with a Force for f, but I just can't find one with velocity and mass alone.

Also, can you please elaborate why M is relevant? I can't see how it'll affect the velocity of m.

Ok, I've been tinkering with it and I am guessing since I'm assuming no friction is involved, I can finally apply the conservation of momentum, where angular momentum is the same anywhere on the circle. Am I right on this?

 

[haruspex]

it seems as if there is some rugged surface

I'd guess that's just staircasing from drawing a freehand curve at low pixel resolution.

why M is relevant?

If the plate has no inertia, as soon as the puck starts to go up the slope the plate will accelerate to match the puck's speed. The puck will never reach the launch point.

angular momentum is the same anywhere on the circle

Circle? The shape of the curve is not given, and is immaterial beyond some basic smoothness requirements.
I don't think angular momentum will be useful here.
Please post/repost your energy and linear momentum equations, taking into account the movement of the plate.

 

[Samei]

Oh, ok. I'll work on it right now. :)

Sorry for taking so long. I had to do some additional reading to make sure of what I was doing.
Here is what I have:
Assuming no friction, KE1 + PE1 = KE2 + PE2
Again, PE1 starts at an arbitrary h=0, so PE1 is 0.
This becomes KE1 = KE2 + PE2
and ((mv1^2)/2) = ((mv2^2)/2) + mgh
(v1^2)/2 = (v2^2)/2 + gh
I can use this later on to solve for either h or v2.

Now, for momentum. Once again, since the surfaces are frictionless, conservation of momentum can be applied. I'm a bit unsure how to do this, but looking at the problem, this is what I have in mind.
mv1 + Mv1 = mv2 + Mv2
It seems a bit counterintuitive. What did I do wrong here?

 

[Samei]

I just realized.
mv1 + Mv1 = mv2 + Mv2 is for collisions.
I'm stuck now.

 

[haruspex]

and ((mv1^2)/2) = ((mv2^2)/2) + mgh

You are still omitting the movement of the plate.

mv1 + Mv1 = mv2 + Mv2

Yes, but be careful about how these velocities relate to your energy equation velocities. In the momentum equation, direction matters.
Might be safest to create separate symbols for the horizontal and vertical components of the puck's initial and final velocities.

mv1 + Mv1 = mv2 + Mv2 is for collisions.

No, it's true whenever there's no external force in the direction of the linear momentum component under consideration.

 

[Samei]

You are still omitting the movement of the plate.

Yes, but be careful about how these velocities relate to your energy equation velocities. In the momentum equation, direction matters.
Might be safest to create separate symbols for the horizontal and vertical components of the puck's initial and final velocities.

No, it's true whenever there's no external force in the direction of the linear momentum component under consideration.

Ok, let me work this out again.
KE1 (of m) + KE1 (of M) + PE1 (of m) = KE2 (of m) + KE2 (of M) + PE2 (of m)
KE1 (of M) does not have initial velocity applied to it, so it zero. Initial height arbitrarily is zero so PE1 is also zero. KE2 (of m) will be at zero velocity at maximum height so this will also be zero.

KE1 (of m) = KE2 (of M) + PE2 (of m)

For momentum, m(v1,x) + M(v1,x) = m(v2,y) + M(v2,x)
where x is the horizontal component and y is the vertical component. Now, I'm curious is (v2,y) is zero here and would that mean my goal is to solve for (v2,x)?

 

[Samei]

I forgot to add: Is v1 for m different from v1 of M? I assumed it for the energy equations since no velocity is applied there but I am not sure if that is valid.

 

[haruspex]

KE1 (of M) does not have initial velocity applied to it,

Right, so why the M v1x term in the momentum?

m(v1,x) + M(v1,x) = m(v2,y) + M(v2,x)

Momentum is a vector. In the vertical direction there is a normal force from the ground, which varies, so you can't use momentum conservation in that direction. Momentum is conserved in the horizontal direction since there's no friction, but v2y doesn't feature in that.
What will be the relationship between the horizontal velocities of puck and plate at the launch point?

 

[Samei]

Thanks for sticking with me on this problem by the way! I'm still working on it, and after discussing this with other people, I found out that if I were to neglect friction, then the bigger plate (M) would not move. Friction is the one that is pushing M to be on the same direction as m. With that in mind, I can say that some KE is being converted from the original m to M. This is described by my energy equations as:
KE of m = KE of M + PE of m
It's basically what I wrote before, but now I have a firm grasp on what's happening. It's a Eureka moment! :)
Or am I completely misunderstanding the laws of physics yet again?

What will be the relationship between the horizontal velocities of puck and plate at the launch point?

I can use the conservation of energy to form this relationship, at least possibly? I know that the velocity of the puck will be greater than the plate, but smaller than initial. That statement may not be directly quantifiable, but it may help.

 

[haruspex]

if I were to neglect friction, then the bigger plate (M) would not move

Wrong. The top of the plate is not horizontal. As soon as the puck starts to go up a slope the normal force is not vertical. This will slow the puck and accelerate the plate.

haruspex said:
What will be the relationship between the horizontal velocities of puck and plate at the launch point?​

I can use the conservation of energy to form this relationship, at least possibly?

No, but you can use geometry. How steep does the slope appear to be at the launch point?

Have another try at the equations in post #8, this time using separate symbols for the horizontal and vertical components of the puck's velocity.

 

[Samei]

No, but you can use geometry. How steep does the slope appear to be at the launch point?

Have another try at the equations in post #8, this time using separate symbols for the horizontal and vertical components of the puck's velocity.

I think I have an idea. There is a reason why the questions suggested that the plate moved vertically.

Here are my updated equations:
Energy: KE1 (of m) = KE2 (of M) + PE2 (of m)
Momentum:
mv1(cos(0)) + Mv1(cos(0)) = mv2(cos(90)) + Mv2(cos(0))
mv1 + Mv1 = Mv2

If all is good, I can work this out.

 

[haruspex]

I think I have an idea. There is a reason why the questions suggested that the plate moved vertically.

Here are my updated equations:
Energy: KE1 (of m) = KE2 (of M) + PE2 (of m)
Momentum:
mv1(cos(0)) + Mv1(cos(0)) = mv2(cos(90)) + Mv2(cos(0))
mv1 + Mv1 = Mv2

If all is good, I can work this out.

No, sorry, both those equations are wrong.
They both assume the puck ends with no horizontal velocity. What makes you think that?

 

[Samei]

No, sorry, both those equations are wrong.
They both assume the puck ends with no horizontal velocity. What makes you think that?

I guess what I really mean was that final vertical velocity would be zero. But how would horizontal velocity play a part in this? I am not exactly sure where to add it for which I can then solve.

Are my equations only lacking?

 

[haruspex]

what I really mean was that final vertical velocity would be zero.

When at its maximum height, yes, but first we have to find the vertical component of the launch speed (i.e. the speed at the time it leaves the plate). Concentrate on that for now.
Your nomenclature for the variables is confusing. Please use these variable names, or something similar:
v = initial (horizontal) speed of puck (given)
v_x=horizontal speed of puck at launch time
v_y=vertical speed of puck at launch time
w=speed of plate at launch time

how would horizontal velocity play a part in this?

The horizontal velocity that the puck retains at launch is taking up some of the KE, so reduces the vertical launch speed.
You didn't answer this question:

How steep does the slope appear to be at the launch time?

When you have answered that correctly you should try to answer this question:

What will be the relationship between the horizontal velocities of puck and plate at the launch time?

 

[Samei]

You didn't answer this question

From the information given and the picture, the slope is as steep as it can be. That means it is about 90 degrees, as it shoots upward. Also, like you said, only the horizontal momentum is conserved. So this is the relationship between the velocities of the puck and plate at launch time: mv = Mw + mv_x

Rewriting the energy equation, at launch:
KE_{1, Puck} = KE_{2, Puck - Horizontal} + KE_{2, Puck - Vertical} + KE_{Plate (Horizontal)} + PE_Puck
I would have three unknowns here, namely h, w, v_x, and v_y.

By direction of components, is it not sufficient that I include the angle (i.e., sin(90) or cos(0))?

 

[haruspex]

That means it is about 90 degrees, as it shoots upward.

Right, but directly upward?

mv = Mw + mvx

Yes, that's the horizontal momentum equation, but it's not the relationship between w and v_x that you can deduce from the vertical slope at the launch point.
What will the relative horizontal velocity be?

I would have three unknowns here, namely h, w, vx, and vy.

Not sure how you are defining h here.
The question says to treat the plate's radius of curvature as much smaller than the jump height. (Alternatively, it could have asked for jump height relative to its initial height, not its height at launch.) Either way, this means we can ignore the gain in PE from initial position to launch point. We just need to find v_y from conservation of KE. We can then find the jump height from that.

 

[Samei]

Right, but directly upward?

No. I didn't realize this before, but some of the horizontal velocity will let it move further.

What will the relative horizontal velocity be?

I am still reluctant, but is the velocity of the plate equal to that of the puck? I've also been using geometry to explain some things, but I don't think it has lead me anywhere. Say, I know that the puck will be perpendicular to the plate at launch.

Energy equation, with minor revision: KE_initial = KE_{puck at launch - horizontal} + KE_{puck at launch - vertical} + KE_{plate at launch}, considering that PE gained is negligible.

 

[haruspex]

No. I didn't realize this before, but some of the horizontal velocity will let it move further.

I am still reluctant, but is the velocity of the plate equal to that of the puck? I've also been using geometry to explain some things, but I don't think it has lead me anywhere. Say, I know that the puck will be perpendicular to the plate at launch.

Energy equation, with minor revision: KE_initial = KE_{puck at launch - horizontal} + KE_{puck at launch - vertical} + KE_{plate at launch}, considering that PE gained is negligible.

Yes, that all looks good. Just bring in the horizontal momentum equation and you have enough to solve it.

 

[Samei]

Yes, that all looks good. Just bring in the horizontal momentum equation and you have enough to solve it.

Ok, I will try it yet again. I can almost see the light at the end of the plate!

Again, thanks for all the help.

 

[Samei]

Yes, that all looks good. Just bring in the horizontal momentum equation and you have enough to solve it.

Just an update, I have equated for it and it all seems to fall into place. This turned out to be more complicated than I initially thought it would be, but I guess it was because I simplified the situation.

 

[insightful]

consider that the puck leaves the plate vertically.

Is it clear that this means vertically in relation to the plate or is it vertically in relation to the table?

 

[haruspex]

Is it clear that this means vertically in relation to the plate or is it vertically in relation to the table?

Are you asking about the question as posed or about Samei's post?

 

[insightful]

Are you asking about the question as posed or about Samei's post?

I referenced "The problem statement." Sorry, I don't understand your question.

 

[haruspex]

I referenced "The problem statement." Sorry, I don't understand your question.

You also need to consider the picture. The only way that the puck could end up moving vertically relative to the table is if the plate curled back at the edge, which it clearly does not.

 

[insightful]

You also need to consider the picture. The only way that the puck could end up moving vertically relative to the table is if the plate curled back at the edge, which it clearly does not.

I did consider the (very crude) picture. I would ask the original source for clarification; just my opinion.

 

[Samei]

I did consider the (very crude) picture. I would ask the original source for clarification; just my opinion.

I'm not quite sure myself, but the teacher does want the students to assume as they will. Looking at the picture, it does not look like it curls.

Just to be safe, I was told to write an observation whenever possible.