Further Mechanics 1 - Oblique Collisions Question

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BrandonS15
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Hello,
Currently, I know that when considering an oblique collision between either a particle colliding obliquely with a barrier/wall or two particles with an oblique collision, that Newton’s experimental law of restitution only applies to the components of velocity that are parallel to the lines of centres or (in the case of a particle and a wall, parallel to the direction of impulse)
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(First and second bullets) Why is this the case? I know its something to do with the direction of impulse but I can’t fully get my head around the concept of impulse determining the direction in which the restitution law applies
I know that in the case of a particle and a wall, the momentum isn’t completely conserved due to the external force acting on the wall which prevents it from moving and if we consider this, a large force to turn the particle in a very short time can be integrated to give change in momentum, impulse. So there we don’t have a closed system therefore momentum isn’t conserved and since impulse gives a measure for how much momentum isn’t conserved then the direction of impulse must decide the direction where momentum isn’t conserved, but what I can’t link is why the restitution law only applies in certain directions when considering oblique collisions.
Any help in understanding intuitively is appreciated, thanks in advance!
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Superfluid123
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Sorry - new to posting latex.
\mathbf{i}+\mathbf{j} and -\mathbf{i}+\mathbf{j}
Here is a computer model of what I am talking about you need to click student preview...
[URL]https://teacher.desmos.com/activitybuilder/custom/5f198611ef1efe1f30feff69[\URL]
Last edited by Superfluid123; 1 year ago
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mqb2766
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These are pretty good
https://www.drfrostmaths.com/resource.php?rid=446
The normal force(s) is applied along the line connecting the centres, just as if one was a wall.
Perpendicular to that line (along the wall), things are unchanged.
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BrandonS15
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(Original post by mqb2766)
These are pretty good
https://www.drfrostmaths.com/resource.php?rid=446
The normal force(s) is applied along the line connecting the centres, just as if one was a wall.
Perpendicular to that line (along the wall), things are unchanged.
I’m still unsure how impulse is the cause for where the restitution law is applied, I have read the powerpoint on the link given, and It doesn’t give a complete understanding as to why we consider restitution in specific directions
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mqb2766
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(Original post by BrandonS15)
I’m still unsure how impulse is the cause for where the restitution law is applied, I have read the powerpoint on the link given, and It doesn’t give a complete understanding as to why we consider restitution in specific directions
Restitution (impulse) is a force, so it acts in a direction. When you're moving in 2d , motion in only one dimension (not necessarily aligned with the axes) is affected. If a ball strikes a wall,motion normal to the wall will be governed by the impulse impact. Motion along the wall is not affected.

You could think about the limiting cases. What happens when it's perpendicular to the wall? What happens when it's parallel to the wall. More generally, you resolve motion into those two components.
Last edited by mqb2766; 1 year ago
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old_engineer
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(Original post by BrandonS15)
Hello,
Currently, I know that when considering an oblique collision between either a particle colliding obliquely with a barrier/wall or two particles with an oblique collision, that Newton’s experimental law of restitution only applies to the components of velocity that are parallel to the lines of centres or (in the case of a particle and a wall, parallel to the direction of impulse)
Name:  650FA852-CC0B-4FB9-BACC-169525628BF1.jpg.jpeg
Views: 57
Size:  27.2 KB
(First and second bullets) Why is this the case? I know its something to do with the direction of impulse but I can’t fully get my head around the concept of impulse determining the direction in which the restitution law applies
I know that in the case of a particle and a wall, the momentum isn’t completely conserved due to the external force acting on the wall which prevents it from moving and if we consider this, a large force to turn the particle in a very short time can be integrated to give change in momentum, impulse. So there we don’t have a closed system therefore momentum isn’t conserved and since impulse gives a measure for how much momentum isn’t conserved then the direction of impulse must decide the direction where momentum isn’t conserved, but what I can’t link is why the restitution law only applies in certain directions when considering oblique collisions.
Any help in understanding intuitively is appreciated, thanks in advance!
Assuming the spheres are smooth, they will be unable to impart any force on one another in the direction tangential to the point of impact. If we then resolve forces / impulses into radial (along the line joining the centres) and tangential (perpendicular to the line joining the centres), then as the tangential component is zero, we can conclude that ALL the force / impulse must be acting along the line joining the centres. It further follows that the components of velocity in the direction perpendicular to the line joining the centres must remain unchanged by the collision.
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BrandonS15
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(Original post by old_engineer)
Assuming the spheres are smooth, they will be unable to impart any force on one another in the direction tangential to the point of impact. If we then resolve forces / impulses into radial (along the line joining the centres) and tangential (perpendicular to the line joining the centres), then as the tangential component is zero, we can conclude that ALL the force / impulse must be acting along the line joining the centres. It further follows that the components of velocity in the direction perpendicular to the line joining the centres must remain unchanged by the collision.
What prevents any force being exerted in the component that is tangent to the point of impact? How do I know that no force is exerted like this so that I can be sure all the impulse is acting parallel to the line of centres?
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old_engineer
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(Original post by BrandonS15)
What prevents any force being exerted in the component that is tangent to the point of impact? How do I know that no force is exerted like this so that I can be sure all the impulse is acting parallel to the line of centres?
If the spheres are smooth, their coefficients of friction must be zero, which in turn implies zero frictional force.
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BrandonS15
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(Original post by old_engineer)
If the spheres are smooth, their coefficients of friction must be zero, which in turn implies zero frictional force.
I undestand that modelling the situation as smooth avoids any frictional forces being involved, but how does this confirm that none of the impulses each sphere receives acts in the tangential direction?
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old_engineer
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(Original post by BrandonS15)
I undestand that modelling the situation as smooth avoids any frictional forces being involved, but how does this confirm that none of the impulses each sphere receives acts in the tangential direction?
An impulse is a force applied for a (usually short) time. No tangential force means no tangential impulse.
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BrandonS15
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(Original post by old_engineer)
An impulse is a force applied for a (usually short) time. No tangential force means no tangential impulse.
Okay that makes further sense and (sorry if you’ve said this already) but how do we know theres no tangential force? I know momentum is conserved in that direction, but don’t see how that completes the explanation..
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old_engineer
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(Original post by BrandonS15)
Okay that makes further sense and (sorry if you’ve said this already) but how do we know theres no tangential force? I know momentum is conserved in that direction, but don’t see how that completes the explanation..
There is no tangential force because the spheres are smooth, with zero coefficient of friction.
Momentum is conserved overall, by the way, not just in the direction perpendicular to the line of centres.
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