Ball Hitting Plank: Find Final Velocity Vf

  • Thread starter Knissp
  • Start date
  • Tags
    Ball
In summary, the conversation discusses a problem involving a plank and a ball, where the final velocity of the ball after impacting the plank is to be found assuming conservation of mechanical energy. The problem also involves calculating the angular momentum and moment of inertia of the system, with some uncertainty about the validity of certain assumptions made. It is found that the final velocity may not be correctly calculated if the rod is assumed to not translate, and using a revised moment of inertia equation may yield a different result.
  • #1
Knissp
75
0

Homework Statement


A plank of length 2L and mass M lies on a frictionless plane. A ball of mass m and speed Vo strikes its end (the plank is standing vertical and the ball strikes the top from the left). Find the final velocity of the ball, Vf, assuming that mechanical energy is conserved and that Vf is along the original line of motion.

Homework Equations


Conservation laws (angular momentum, energy)

The Attempt at a Solution


I am having conceptual difficulty here. I know that initially angular momentum (about the center of the rod) is given by [tex]\widehat{L_0} = m V_0 L[/tex]. After impact, angular momentum is [tex]\widehat{L_f} = m V_f L + I \omega [/tex]. Conservation of energy gives [tex]1/2 m V_0^2 = 1/2 m V_f^2 + 1/2 I \omega^2[/tex]. Given I, I can solve for [tex]\omega[/tex] and substitute and do some algebra to find [tex]V_f[/tex] (which I have already done but don't want to type; it's not relevant to my question). I used [tex]I = 1/12 M (2L)^2 = 1/3 M L^2[/tex], which is the moment of inertia about the center of the rod. I am not sure, however, if it is valid to assume that the rod rotates about its center. I feel as if this is intuitively true but am not sure how to completely convince myself of this, so perhaps I am incorrect. Also, I'm assuming that the rod does not translate. Is this a valid assumption? Does anyone have any input?EDIT: The hint for this question says if m=M, then Vf = 3/5 Vo. The technique I used does not yield that result, so I must be setting this up incorrectly. This tells me that at least one of my assumptions is flawed. Any ideas on which one?
 

Attachments

  • phys.bmp
    87.3 KB · Views: 572
Last edited:
Physics news on Phys.org
  • #2
EDIT 2: I think the mistake was that I assumed the rod did not translate, which it probably does. I think this means that I should have used I = 1/3 (M+m) L^2 instead of I = 1/3 ML^2. Does anyone agree?
 
  • #3


Your approach using conservation of angular momentum and energy is correct. However, there are a few things to consider in this problem:

1. Is the ball sticking to the plank or bouncing off? This will affect the final velocity.

2. The moment of inertia for a rod rotating about its center is indeed 1/12 ML^2, but in this problem, the rod is rotating about its end. The moment of inertia for a rod rotating about one end is 1/3 ML^2. This is because the mass is distributed differently in this case.

3. In this problem, the rod does not translate, so you can assume that the center of mass of the rod remains stationary.

Taking these factors into account, you can set up the conservation equations as follows:

Initial angular momentum = mVoL
Final angular momentum = mVfL + (1/3)ML^2ω

Initial kinetic energy = (1/2)mVo^2
Final kinetic energy = (1/2)mVf^2 + (1/2)(1/3)ML^2ω^2

From here, you can solve for Vf and ω and substitute back into the equations to get the final velocity along the original line of motion.

Hope this helps!
 

FAQ: Ball Hitting Plank: Find Final Velocity Vf

1. What is the equation for finding final velocity (Vf) in a ball hitting plank scenario?

The equation for finding final velocity in this scenario is Vf = √(Vi^2 + 2ad), where Vf is final velocity, Vi is initial velocity, a is acceleration, and d is distance.

2. How does the mass of the ball affect the final velocity in this scenario?

The mass of the ball does not directly affect the final velocity in this scenario. However, it can affect the acceleration of the ball and therefore impact the final velocity.

3. Can the angle at which the ball hits the plank affect the final velocity?

Yes, the angle at which the ball hits the plank can affect the final velocity. A steeper angle will result in a higher final velocity, while a shallower angle will result in a lower final velocity.

4. Is there a maximum final velocity that can be achieved in this scenario?

Yes, there is a maximum final velocity that can be achieved in this scenario. It is determined by the initial velocity, acceleration, and distance.

5. How does the material of the plank affect the final velocity?

The material of the plank can affect the final velocity in terms of the force of impact and the amount of energy absorbed. A softer material may result in a lower final velocity, while a harder material may result in a higher final velocity.

Back
Top