Coefficient of restitution of a particle

In summary, when a particle with initial velocity of (3i + 5j) m-s−1 collides with a smooth plane wall at an unknown angle, resulting in a final velocity of (-2i-j) m-s−1, the coefficient of restitution can be calculated by finding the direction of the surface normal, which is (-5i - 6j). Using the components of the velocities about the line of impact, the ratio of the velocities after and before the impact is 16/45, giving the coefficient of restitution as 16/45.
  • #1
erisedk
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Homework Statement


A particle moving with initial velocity vi = (3i + 5j) m-s−1, collides with a smooth plane wall placed at some orientation to the particle’s trajectory. The resulting velocity of the particle is vf = (−2i − j) m-s−1. The coefficient of restitution for this collision is
Ans: 16/45

Homework Equations


e = velocity of separation/velocity of approach

The Attempt at a Solution


So, my first attempt was
e = [ 0 - (−2i − j) ]/ [ (3i + 5j) - 0 ] = (2i + j)/(3i +5j)
I thought of taking the speeds then, which gave me e = √5 / √34
Which is obviously not the answer.
Then I tried to google COR, and i got-- "representing the ratio of speeds after and before an impact, taken along the line of the impact"

I have a feeling that my answer is wrong because I'm not incorporating "line of impact". But I have no idea how to, because I'm not given the angle at which the particle collides with the wall.
 
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  • #2
There is an underlying assumption that the wall can only provide a force (and thus acceleration) perpendicular to itself (the keyword is "smooth"). You can use this to find out how the wall is tilted.
 
  • #3
Could you please elaborate a bit more?
 
  • #4
Are you able to compute the direction of the velocity change? The force is normal to the surface, so the velocity change is going to be in the normal direction of the surface and so computing the velocity change will give you the direction of the surface normal.
 
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  • #5
Ok thank you!
I got it.
vf - vi = (-5i - 6j) = direction of surface normal
Taking the components of velocities about line of impact, vi = (-5i - 6j) . (3i + 5j) = 45 & vf = (-5i - 6j) . ( -2i-j ) = 16
Hence, e = 16/45.
 
  • #6
Yes, just one comment: What you really computed was not the velocity components in the normal direction, but the magnitude of the projections of the velocities on a normal that was not normalised. Now, for this problem it does not matter as you are computing the ratio between two components and the normalisation cancels out. However, in general, you might want to normalise the surface normal when computing the component normal to the surface.
 
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What is the coefficient of restitution of a particle?

The coefficient of restitution (COR) of a particle is a measure of the elasticity of a collision between two objects. It is defined as the ratio of the relative velocity of separation after a collision to the relative velocity of approach before the collision.

How is the coefficient of restitution of a particle calculated?

The coefficient of restitution can be calculated by measuring the initial velocity and final velocity of the particle after a collision and plugging those values into the equation COR = (final velocity / initial velocity).

What does a high coefficient of restitution indicate?

A high coefficient of restitution indicates that the collision between two particles is highly elastic, meaning that there is minimal loss of kinetic energy during the collision. This is often seen in collisions between two hard objects, such as two billiard balls.

What factors can affect the coefficient of restitution of a particle?

The coefficient of restitution can be affected by a variety of factors, including the material properties of the two objects (such as their elasticity and surface roughness), the angle of collision, and the relative masses and velocities of the particles.

Why is the coefficient of restitution important in physics and engineering?

The coefficient of restitution is an important concept in physics and engineering because it helps us understand and predict the behavior of objects during collisions. It is especially useful in designing and analyzing systems involving high-speed impacts, such as in sports equipment or car crashes.

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