# Maximum height a block reaches after separating from a curved moving ramp

• ItsukaKitto
In summary: I think you would need to test that.If the lower block manages to reach the place where the curved surface becomes purely vertical then yes, the horizontal velocity of both blocks will be the same.
ItsukaKitto
Homework Statement
A body of mass M with a small block of mass m placed on it, rests on a smooth horizontal plane. The surface of the body M is horizontal near the smaller mass and gradually curves to become vertical. The block is set in motion in the horizontal direction with a velocity v. To what height, relative to the initial level, will the block rise after breaking off the body M? Friction is assumed to be absent.
Relevant Equations
F = ma, momentum conservation
Conservation of energy
Total work = ∆K.E.
Diagram attached at the endI personally think there's something wrong with this question, and I'd like if someone can tell me whether it's the question that is wrong or my approach.

If I attempt the solution thinking that M should be stationary, the solution is simple. 0 - 1/2 mv^2 = -mgh, gives h = v^2/(2g). However, this is not the right answer, so I assumed the lower block moves (because this problem was given in a section titled "momentum" ), but for mass M to move, there should be a force on it...which can only arise if there is friction.

If I assume friction is present, I get the velocity
w = mv/(M+m) as the common velocity for m and M.

And to obtain the height, I can use the work energy theorem:
>>> W(gravity) + W(friction) = 1/2 m(v1)^2 + 1/2Mw^2 - 1/2mv^2
note: here v1 is the velocity of the block as it breaks off with M.

I found that W(friction) = -v^2(Mm)/2(M+m)
So substituting and setting v1 = w, (as I think, the horizontal component when the block is breaking off, wouldn't change) I get:
>>>2*W(gravity) = w^2(m + M) - mv^2 + v^2(Mm)/(M+m)

>>>-2mgh = v^2(m^2)/(m+M) - mv^2 + v^2 (Mm)/(M+m)

>>> Meaning that h is precisely zero...(is it?)

When I checked the answer to this problem, it was actually Mv^2/2g(m+M) and I found out a way they could have arrived at this:
Using conservation of energy, and ignoring the work done by friction:
1/2 mv^2 = 1/2 m(v1)^2 + 1/2 Mw^2 + mgh
Now setting v1 = w, we get:
mv^2 - (m+M) w^2 = 2mgh

>> h = Mv^2 /2g(M+m)

I don't think this approach is right, because if they are considering friction, why are they ignoring the work it does?

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ItsukaKitto said:
but for mass M to move, there should be a force on it...which can only arise if there is friction.
There is also the normal force. If you assume that the small block has enough initial velocity to get past the curved section then you have enough information to calculate what you need to know.

ItsukaKitto
jbriggs444 said:
There is also the normal force. If you assume that the small block has enough initial velocity to get past the curved section then you have enough information to calculate what you need to know.
Thanks for replying, I've a few questions...
It's not possible for the normal force to cause the lower block to have a horizontal velocity, right?
Do you think the "most accurate" answer should actually be v^2/2g then, assuming no friction?
In saying that the block should have enough initial velocity to get past the curved section, are you referring to the case where friction is present?

ItsukaKitto said:
It's not possible for the normal force to cause the lower block to have a horizontal velocity, right?
Why not? In what directions does it act?

jbriggs444 said:
Why not? In what directions does it act?
In the curved part, the normal force would have a horizontal component, and the lower block will have a constant velocity in the horizontal direction...then I suppose I can use momentum conservation to find it. I'm not sure however, will the horizontal velocity of both the blocks be the same then? I see if they're stuck together till the end that must be true, should I just assume this?

ItsukaKitto said:
In the curved part, the normal force would have a horizontal component, and the lower block will have a constant velocity in the horizontal direction...then I suppose I can use momentum conservation to find it. I'm not sure however, will the horizontal velocity of both the blocks be the same then? I see if they're stuck together till the end that must be true, should I just assume this?
You are told that there is no friction. This is a useful piece of information. It let's you know that you can use momentum conservation. The total momentum of lower plus upper block (in the horizontal direction at least) must be conserved because no external force has a horizontal component. As you suspected.

If the lower block manages to reach the place where the curved surface becomes purely vertical then yes, the horizontal velocity of both blocks will be the same.

jbriggs444 said:
You are told that there is no friction. This is a useful piece of information. It let's you know that you can use momentum conservation. The total momentum of lower plus upper block (in the horizontal direction at least) must be conserved because no external force has a horizontal component. As you suspected.

If the lower block manages to reach the place where the curved surface becomes purely vertical then yes, the horizontal velocity of both blocks will be the same.
I see, so would I have to assume the blocks stick together through the entire time the smaller one is on the curved part? Most likely they would have different velocities in that part, and I think whether they will stick together here, depends on the geometry of the big block too. Is this true?

ItsukaKitto said:
I see, so would I have to assume the blocks stick together through the entire time the smaller one is on the curved part? Most likely they would have different velocities in that part, and I think whether they will stick together here, depends on the geometry of the big block too. Is this true?
It is difficult to imagine them flying apart since the curve is concave throughout.

jbriggs444 said:
It is difficult to imagine them flying apart since the curve is concave throughout.
For the sake of conserving momentum here, I think that's a reasonable assumption. Thank you for your help.

## 1. What is the maximum height a block can reach after separating from a curved moving ramp?

The maximum height a block can reach after separating from a curved moving ramp depends on several factors, such as the initial velocity of the block, the angle of the ramp, and the height of the ramp. It can be calculated using the equations of motion and the principles of conservation of energy.

## 2. How does the angle of the ramp affect the maximum height reached by the block?

The angle of the ramp plays a crucial role in determining the maximum height reached by the block. A steeper ramp will result in a higher maximum height, as the block will have a greater initial velocity and will travel a longer distance before reaching the ground. However, if the angle is too steep, the block may not have enough time to separate from the ramp and will not reach its maximum height.

## 3. Can the maximum height reached by the block be greater than the height of the ramp?

Yes, it is possible for the maximum height reached by the block to be greater than the height of the ramp. This can occur if the initial velocity of the block is high enough and the angle of the ramp is steep enough. In this case, the block will continue to travel upwards even after separating from the ramp, reaching a height greater than the height of the ramp.

## 4. How does the initial velocity of the block affect the maximum height reached?

The initial velocity of the block has a direct impact on the maximum height it can reach after separating from the ramp. A higher initial velocity will result in a higher maximum height, as the block will have more kinetic energy to convert into potential energy. However, if the initial velocity is too low, the block may not have enough energy to reach its maximum height and will fall back to the ground sooner.

## 5. Is there a limit to the maximum height a block can reach after separating from a curved moving ramp?

There is no theoretical limit to the maximum height a block can reach after separating from a curved moving ramp. However, in practical scenarios, factors such as air resistance and friction will limit the maximum height that can be achieved. Additionally, if the ramp is too short or the initial velocity is too low, the block may not have enough energy to reach a significant height.

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