Math Help - Vector problem

1. Vector problem

A particle collides with a surface, at velocity vector $v$. The normal vector for the point at which is collides is vector $n$. $\hat v$ and $\hat n$ are the unit vectors for v and a respectively.

1) Using the fact that angle of incident = angle of reflection, what is the new vector $o$, which represents the outgoing velocity, assuming there is no loss of energy.

2) Given a coefficient of restitution, c, how would the vector $o$ differ?

2. Part c) If friction, with coefficient f, is introduced, how would this affect the outgoing vector, o?

3. Originally Posted by scorpion007
A particle collides with a surface, at velocity vector $v$. The normal vector for the point at which is collides is vector $n$. $\hat v$ and $\hat n$ are the unit vectors for v and a respectively.

1) Using the fact that angle of incident = angle of reflection, what is the new vector $o$, which represents the outgoing velocity, assuming there is no loss of energy.
The component of $\bold{v}$ normal to the surface has
its sign reversed:

$
\bold{o}=\bold{v}-2 (\bold{v}.\bold{\hat{u}}) \bold{\hat{u}}
$

RonL

4. Originally Posted by scorpion007
A particle collides with a surface, at velocity vector $v$. The normal vector for the point at which is collides is vector $n$. $\hat v$ and $\hat n$ are the unit vectors for v and a respectively.

2) Given a coefficient of restitution, c, how would the vector $o$ differ?
Remove component of $\bold{v}$ normal to the surface and replace with $-c$ times it:

$
\bold{o}=\bold{v}-(1+c) (\bold{v}.\bold{\hat{u}}) \bold{\hat{u}}
$

RonL

5. Originally Posted by scorpion007
Part c) If friction, with coefficient f, is introduced, how would this affect the outgoing vector, o?
It would apply an additional change in component parallel to the surface
proportional to $f\ dt$ where $dt$ is the duration of
contact. The constant of proportionality would depend on factors we are not
told (or a definition of coefficient of friction of which I am unaware).

RonL