Elastic Angular Momentum problem

In summary: If so, then your equations are almost right, but the v0 on the right side should be squared, not just v0.Also, as mentioned in the previous summary, the mass should not be indexed and the angles should be in radians. In summary, to show that in an elastic collision between two identical billiard balls, if one ball strikes the other at rest and is deflected by 45 degrees, then the other ball must move at 90 degrees to the first and with the same speed. This can be proved by applying the conservation of momentum and kinetic energy equations, where the incoming ball is deflected by θ1=π/4 from the x-axis and the other ball makes an angle of π/2
  • #1
Violagirl
114
0

Homework Statement


A billiard ball strikes an identical billiard ball initially at rest and is deflected 45 degrees from its original position. Show that if the collision is elastic, the other ball must move at 90 degrees to the first and with the same speed.

Homework Equations



Momentum:
mv = m1v1 + m2v2

Kinetic energy = 1/2m1v12 =1/2m1v12 + m2v22

The Attempt at a Solution



I think I've come up with three equations so far in breaking the problem up but I have no idea if I'm the right track or not. Any input is much appreciated.

1. Pfx=Pix
vxcos 45° + m2vxcos θ = v0

2. Pfy=Piy

vysin 45° - vysin θ = 0
vysin 45°=vysin θ

3. 1/2v02 = v12+v22
 
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  • #2
Some observations:
If ##v_x## referrs to the x-component of velocity ##v## then why have you got it paired with trig functions?

If the balls are identical, then there is no need to index mass like that ##m_2## and that mass does not belong in the relation anyway.

If the initial momentum is ##m\vec{v}=mv\hat{\imath}## ...
And the final momentum is ##m\vec{v}_1+m\vec{v}_2## then you want the components in terms of ##v_1## and ##v_2##.

I also think you should be doing angles in radiens (get used to it).
Note: ##\sin(\pi/4)=\cos(\pi/4)=1/\sqrt{2}##
 
  • #3
Thank you for your input, I'll have to back and reevaluate the equations I found. I set it up the way I did with the hope of eliminating and plugging unknowns into equations. I honestly am confused though on to best go about solving this problem...=/
 
  • #4
All conservation problems are worked pretty much the same way - you write a heading "before" and sketch the situation. Then write the equations that describe what you drew.
Do the same for "after". Then state the conservation laws and apply them.

In this case:
before (one mass incoming along the x axis)
##p_x=mv##
##p_y=0##
##K=\frac{1}{2}mv^2##

after (the incoming mass is deflected by θ1=π/4 from the x axis.[*])
##p_x=v_1\cos(\theta_1)+v_2\cos(\theta_2)##
##p_y=\cdots##
##K=\cdots##

conservation of momentum[\b]
##p_{x,before} = p_{x,after}##
##p_{y,before}\cdots##

conservation of kinetic energy (elastic collision)
##K_{before}\cdots##

------------------------

[*] the trig depends on the geometry.
Note: if the first (incoming) ball is deflected by π/4, and the other ball makes an angle of π/2 to the first - then what angle does it make to the x-axis?
 
  • #5
Violagirl said:
vxcos 45° + m2vxcos θ = v0
Do you mean
v1cos 45° + v2cos θ = v0
where v1 and v2 are the speeds of the balls after collision?
 

What is Elastic Angular Momentum problem?

The Elastic Angular Momentum problem is a physics problem that involves calculating the angular momentum of an object or system before and after an elastic collision. It is commonly encountered in mechanics and can be used to analyze the conservation of angular momentum in a closed system.

What are the key concepts involved in solving an Elastic Angular Momentum problem?

The key concepts involved in solving an Elastic Angular Momentum problem include the conservation of angular momentum, elastic collisions, and the principle of superposition. It is also important to have a clear understanding of the physical properties of the objects involved, such as their mass, velocity, and moment of inertia.

How do you calculate the angular momentum in an Elastic Angular Momentum problem?

The angular momentum in an Elastic Angular Momentum problem can be calculated using the formula L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. In cases where the moment of inertia is not constant, the formula can be written as L = mr²ω, where m is the mass of the object and r is its distance from the axis of rotation.

What is the principle of conservation of angular momentum?

The principle of conservation of angular momentum states that the total angular momentum of a closed system remains constant, before and after an interaction or collision. This means that the initial angular momentum of the system is equal to the final angular momentum, and no external torque is acting on the system.

What are some real-life examples of Elastic Angular Momentum problems?

Some real-life examples of Elastic Angular Momentum problems include the swinging of a pendulum, the rotation of a spinning top, and the motion of a gymnast performing a somersault in the air. These situations involve the conservation of angular momentum and can be analyzed using the principles of mechanics.

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