Finding Coefficient of Restitution for 2D Collision

[bobred]

Linear momentum 2d collision

Homework Statement

Two particles A and B of mass $$m$$ and $$3m$$ respectively, A collides with B. Find the coefficient of restitution $$e$$ if $$\textbf{v}_{A}$$ is purely in the $$\textbf{j}$$-direction.

Velocity of each particle before collision.
$$\dot{\textbf{r}}_{A}=9 \textbf{i}+5\textbf{j}$$
$$\dot{\textbf{r}}_{B}=2 \textbf{i}+2\textbf{j}$$

The x and y velocity components before collision
$$\dot{x}_{A}$$, $$\dot{y}_{A}$$, $$\dot{x}_{B}$$ and $$\dot{y}_{B}$$

The x and y velocity components after collision
$$\dot{X}_{A}$$, $$\dot{Y}_{A}$$, $$\dot{X}_{B}$$ and $$\dot{Y}_{B}$$

Homework Equations

The common tangent is the $$\textbf{j}$$ axis.

$$\dot{y}_{A}\textbf{j}=\dot{Y}_{A}\textbf{j}$$ and $$\dot{y}_{B}\textbf{j}=\dot{Y}_{B}\textbf{j}$$

$$(\dot{X}_{A}-\dot{X}_{B})\textbf{i}=-e(\dot{x}_{A}-\dot{x}_{B})\textbf{i}$$

$$m\dot{\textbf{r}}=m\textbf{v}$$

The Attempt at a Solution

With the values above I find the velocities after collision

$$\textbf{v}_{A}=(-\frac{21}{4}e+\frac{15}{4})\textbf{i}+5\textbf{j}$$
$$\textbf{v}_{B}=(\frac{7}{4}e+\frac{15}{4})\textbf{i}+2\textbf{j}$$

How do I find $$e$$ if $$\textbf{v}_{A}$$ is purely in the $$\textbf{j}$$-direction?

If I use(which is in the i-direction)

$$(\dot{X}_{A}-\dot{X}_{B})=-e(\dot{x}_{A}-\dot{x}_{B})$$

I get $$e=0$$, a totally inelastic collision.

Thanks in advance.

 

[kuruman]

What do you think is the correct expression for the coefficient of restitution when you have a collision in 2-d? How did you conclude that the y-component of each particle's momentum is the same before and after the collision? Finally, what is a "common tangent" and why is it the y-axis?

 

[bobred]

Hi

The common tangent is the plane perpendicular to the common normal plane. We are told the common normal is in the i-direction. In the course text we are told

'In an (instantaneous) collision between two smooth non-rotating objects, where the area of contact at the moment of impact lies on a common tangent plane, the velocities parallel to the tangent plane remain unchanged before and after impact.'

Thanks

 

[kuruman]

Hi

The common tangent is the plane perpendicular to the common normal plane. We are told the common normal is in the i-direction. In the course text we are told

'In an (instantaneous) collision between two smooth non-rotating objects, where the area of contact at the moment of impact lies on a common tangent plane, the velocities parallel to the tangent plane remain unchanged before and after impact.'

Thanks

OK, then, the coefficient of restitution is given by

$$e=\left| \frac{\dot{X_A}-\dot{X}_B}{\dot{x_A}-\dot{x}_B} \right|$$
You know both terms in the denominator and it is given that

$$\dot{X}_A=0$$

Use momentum conservation in the x-direction to find the remaining term.

 

[bobred]

Hi

Thanks for the reply, in explaining to you the coordinate system it got me thinking again and obviously there is no $$\dot{X}_A$$ so obviously I need to solve

$$(\dot{X}_{A}-\dot{X}_{B})=-e(\dot{x}_{A}-\dot{x}_{B})$$

for e

Thanks

 

[kuruman]

I hope you can finish from this point.

 

[bobred]

Yes, all done, thanks again