# A complicated problem of motion on rough surfaces

• Sihas N

#### Sihas N

Homework Statement
The sphere A is released from the top of a rough inclined plane with an inclination of \mathbit{\alpha} to the horizontal. Let the length of the inclined plane be \mathbit{d}_\mathbf{1}. After a time interval ∆t1, another sphere B is released from the top with an initial velocity \mathbit{v}_\mathbf{1} and simultaneously a third sphere C is also projected vertically upwards the plane with an initial velocity \mathbit{v}_\mathbf{2} \left(\mathbit{v}_\mathbf{2}>\mathbit{v}_\mathbf{1}\right). Both spheres B and C collide with the sphere A at the same instant. The sphere B then rebounds instantaneously for a distance ∆d upwards the vertical plane and the sphere C immediately stops at the moment of collision. The inclination of the plane becomes \mathbit{\beta} at the instance of collision \left(\mathbit{\alpha}>\mathbit{\beta}\right). The time interval between the stoppage and the restart of the motion is ∆t2. Then all three spheres start to roll down the plane. All three spheres are made of different materials. The coefficient of friction between the plane and the sphere A, B and C is \mathbit{\mu}_\mathbf{1},\ \mathbit{\mu}_\mathbf{2} and \mathbit{\mu}_\mathbf{3} respectively. The coefficient of restitution between the sphere A and the sphere B is \mathbit{e}_\mathbf{1} and between the sphere A and sphere C is \mathbit{e}_\mathbf{2}. Then all three spheres reach a rough horizontal floor where the coefficient of friction between the surface and the sphere A, B and C is \mathbit{\mu}_\mathbf{4},\ \mathbit{\mu}_\mathbf{5} and \mathbit{\mu}_\mathbf{6} respectively. Spheres A and C come to rest keeping distances between each other as \mathbit{l} where the sphere C comes to rest at the base of another inclined plane, whereas the sphere B keeps moving with a uniform velocity which it attains after the collision. The distance between the first inclined plane and the second inclined plane is \mathbit{d}_\mathbf{2} Once they come to the horizontal floor, an impulsive force of I is applied on to the sphere B. It moves with an acceleration \mathbit{f} and collides again with the sphere A and stops. Thereafter, the sphere A too moves with a uniform velocity \mathbit{v}_\mathbf{3} and then collides again with the sphere C. As soon as the sphere C is collided, it is moved upwards the rough plane inclined at an angle \mathbit{\theta} to the horizontal where the coefficient of friction between the plane and the sphere is \mathbit{\mu}_\mathbf{7}. But the sphere C could only move upwards for a distance \mathbit{d}_\mathbf{3} of the total length of the inclined plane. Then it rolls back down and re-collides with the sphere A and it too moves backwards with a velocity \mathbit{v}_\mathbf{4} but stops just touching the sphere B, such that none of the spheres move thereafter. Assume that the sphere C stops immediately after colliding with the sphere A. This whole motion takes a time \mathbit{T}. Let the masses of the spheres A, B and C be \mathbit{m}_\mathbf{1},\ \mathbit{m}_\mathbf{2} and \mathbit{m}_\mathbf{3} respectively. Considering the motion from the first inclined plane to the other inclined plane as positive draw the necessary VT graph for the motion of three spheres until they all come to rest.
Relevant Equations
Since the question just asks to draw the necessary VT graph, I did not involve any equations.
I couldn't draw the motion after the collision, since the whole angular displacement of the plane got me confused.

Here's an attempt to fix up the LaTeX….
(I read the "\mathbit" is a bug in some Word software.)

The sphere A is released from the top of a rough inclined plane with an inclination of ##{\alpha} ## to the horizontal. Let the length of the inclined plane be ##{d}_\mathbf{1}##. After a time interval ∆t1, another sphere B is released from the top with an initial velocity ## {v}_\mathbf{1}## and simultaneously a third sphere C is also projected vertically upwards the plane with an initial velocity ##{v}_\mathbf{2} \left({v}_\mathbf{2}>{v}_\mathbf{1}\right)##. Both spheres B and C collide with the sphere A at the same instant.

The sphere B then rebounds instantaneously for a distance ∆d upwards the vertical plane and the sphere C immediately stops at the moment of collision. The inclination of the plane becomes ##{\beta}## at the instance of collision ##\left({\alpha}>{\beta}\right)##. The time interval between the stoppage and the restart of the motion is ∆t2.
Then all three spheres start to roll down the plane. All three spheres are made of different materials. The coefficient of friction between the plane and the sphere A, B and C is ##{\mu}_\mathbf{1},\ {\mu}_\mathbf{2}## and ##{\mu}_\mathbf{3}## respectively.

The coefficient of restitution between the sphere A and the sphere B is ##{e}_\mathbf{1}## and between the sphere A and sphere C is ##{e}_\mathbf{2}##.

Then all three spheres reach a rough horizontal floor where the coefficient of friction between the surface and the sphere A, B and C is ##{\mu}_\mathbf{4},\ {\mu}_\mathbf{5}## and ##{\mu}_\mathbf{6}## respectively.

Spheres A and C come to rest keeping distances between each other as ##{l}## where the sphere C comes to rest at the base of another inclined plane, whereas the sphere B keeps moving with a uniform velocity which it attains after the collision. The distance between the first inclined plane and the second inclined plane is ##{d}_\mathbf{2} ##.
Once they come to the horizontal floor, an impulsive force of I is applied on to the sphere B. It moves with an acceleration ##{f}## and collides again with the sphere A and stops. Thereafter, the sphere A too moves with a uniform velocity ##{v}_\mathbf{3} ## and then collides again with the sphere C.

As soon as the sphere C is collided, it is moved upwards the rough plane inclined at an angle ##{\theta} ## to the horizontal where the coefficient of friction between the plane and the sphere is ##{\mu}_\mathbf{7}##. But the sphere C could only move upwards for a distance ##{d}_\mathbf{3}## of the total length of the inclined plane. Then it rolls back down and re-collides with the sphere A and it too moves backwards with a velocity ##{v}_\mathbf{4}## but stops just touching the sphere B, such that none of the spheres move thereafter.

Assume that the sphere C stops immediately after colliding with the sphere A. This whole motion takes a time ##{T}##. Let the masses of the spheres A, B and C be ##{m}_\mathbf{1},\ {m}_\mathbf{2}## and ##{m}_\mathbf{3} ## respectively.

Considering the motion from the first inclined plane to the other inclined plane as positive draw the necessary VT graph for the motion of three spheres until they all come to rest.

Since the question just asks to draw the necessary VT graph, I did not involve any equations.

I couldn't draw the motion after the collision, since the whole angular displacement of the plane got me confused.

berkeman and kuruman
First, I with to thank @haruspex for making the post more legible although not more coherent.

Second, I wish to ask @Sihas N whether this is a homework problem assigned in a course and, if so, what kind of course? I am asking because it does not look like a professionally designed question intended to teach you something. It looks like a gotcha question designed by an amateur with the intention to confuse you, which it did.

This isn't actually any problem related to a college course. Just a semester practice exam problem from my high school. I have the enough knowledge to understand each of the stages in the question. But putting it all together gives me a headache.

I have the enough knowledge to understand each of the stages in the question.
I'm not sure I do.

It's a rough plane, B is sent down it at some speed and C is "projected" up it. So are they initially rolling or sliding? If sliding, we need to be able to work out whether they transition to rolling before impact, so we’ll have to assume they are rolled.

We are told the spheres are of different materials, but not their relative sizes, masses or densities, so that's useless information.

There is no such thing as simultaneity. One collision will happen slightly before the other, and the order makes a big difference.

Since the spheres are rotating, and we are not told a coefficient of friction between them, we'll have to assume there is none.

I have no idea what Δt2 is doing. It sounds like the whole system is frozen for that period, so why do we care about it?

If the ramp angle suddenly reduces, what happens to the spheres? Are they now in free fall, or are we to take them as being somehow anchored to the plane, magnetically perhaps?

I believe the whole question is a practical joke.

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nasu and berkeman
Just a semester practice exam problem from my high school.
Then you probably have additional practice exam problems and have sat through actual exams. Was what you saw similar to this?
I believe the whole question is a practical joke.
Ditto.

Then you probably have additional practice exam problems and have sat through actual exams. Was what you saw similar to this?

Ditto.
Thankfully actual exam questions are not that crazy. But sometimes teachers think that giving similar questions will "enlighten" our analytical thinking. Anyhow thanks for even reading the question.

I'm not sure I do.

It's a rough plane, B is sent down it at some speed and C is "projected" up it. So are they initially rolling or sliding? If sliding, we need to be able to work out whether they transition to rolling before impact, so we’ll have to assume they are rolled.

We are told the spheres are of different materials, but not their relative sizes, masses or densities, so that's useless information.

There is no such thing as simultaneity. One collision will happen slightly before the other, and the order makes a big difference.

Since the spheres are rotating, and we are not told a coefficient of friction between them, we'll have to assume there is none.

I have no idea what Δt2 is doing. It sounds like the whole system is frozen for that period, so why do we care about it?

If the ramp angle suddenly reduces, what happens to the spheres? Are they now in free fall, or are we to take them as being somehow anchored to the plane, magnetically perhaps?

I believe the whole question is a practical joke.
Now only I realized that I actually haven't understood any. I feel like I just wasted your time with this crazy question. Thanks for answering though.

berkeman