How Does a Ball's Mass Affect Its Final Velocity After Colliding with a Plank?

[WannabeNewton]

Homework Statement

A plank of length [itex]2l[/itex] and mass M lies on a friction-less plane. A ball of mass m and speed [itex]v_{0}[/itex] strikes its end as shown. Find the final velocity [itex]v_{f}[/itex] of the ball assuming mechanical energy is conserved and [itex]v_{f}[/itex] is along the original line of motion. Ans. clue. For m = M, [itex]v_{f} = \frac{3}{5}v_{0}[/itex].

Homework Equations

The Attempt at a Solution

Ok so because [itex]\sum F = \partial _{t}P = 0[/itex] and [itex]\sum \tau = \partial _{t}L = 0[/itex] (internal collision forces cancel as per Newton's 3rd law) we can use conservation of both linear and angular momentum. We are free to choose a viable coordinate system so any coordinate system with its origin fixed on the line of motion of the point particle of mass m will do the trick. In this frame, [itex]L_{0} = 0[/itex] because the plank is initially at rest and the particle's momentum is directed parallel to its position vector with respect to the origin in this frame. Since the question states that even after the collision the particle moves along the same line of motion, the angular momentum after comes entirely from the plank which is now rotating as well as translating in some form so [itex]L_{f} = I_{com}\omega(-\hat{k}) + lMV(\hat{k}) = (lMV - \frac{1}{3}Ml^{2}\omega)\hat{k} = 0 [/itex] and we get that [itex]\omega = \frac{3V}{l}[/itex]. We also have conservation of linear momentum and here is where I got confused. I'll do as I did on my note book and state my concern after: [itex]mv_{0} = -mv_{f} + MV[/itex] so [itex]V^{2} = \frac{m^{2}}{M^{2}}(v_{0} + v_{f})^{2}[/itex] and using this together with the expression for the angular velocity of the plank after the collision I was able to simply the conservation of mechanical energy equation, [itex]\frac{1}{2}mv_{0}^{2} = \frac{1}{2}mv_{f}^{2} + \frac{1}{2}MV^{2} + \frac{1}{2}I_{com}w^{2}[/itex], down to [itex](1 + \frac{4m}{M})v_{f}^{2} + 8\frac{m}{M}v_{0}v_{f} + (\frac{4m}{M} - 1)v_{0}^{2} = 0[/itex] which gives the solutions [itex]v_{f} = -{v_{0}\frac{(\frac{4m}{M} \pm 1)}{(\frac{4m}{M} + 1)}}[/itex]. If I use m = M for the Ans. clue. then I get [itex]v_{f} = -\frac{3}{5}v_{0}[/itex] or [itex]v_{f} = -v_{0}[/itex]. My questions are these: I assumed at the beggining that the point particle would shoot backwards after colliding with the plank hence the negative sign but if I assume it keeps going in the same direction and do everything up till now the exact same way I get the right answer [itex]v_{f} = \frac{3}{5}v_{0}[/itex] but it makes intuitive sense to me that the particle would bounce back after hitting the plank thus causing the plank to rotate, and translate in the opposite direction; why would the particle keep moving forward? Secondly, there is the case v0 = vf and I don't see a justification for why this solution can be eliminated other than the fact that it would be a trivial solution where nothing at all happens to the plank. Thank you.

 

[ehild]

I assumed at the beggining that the point particle would shoot backwards after colliding with the plank hence the negative sign but if I assume it keeps going in the same direction and do everything up till now the exact same way I get the right answer [itex]v_{f} = \frac{3}{5}v_{0}[/itex] but it makes intuitive sense to me that the particle would bounce back after hitting the plank thus causing the plank to rotate, and translate in the opposite direction; why would the particle keep moving forward? Secondly, there is the case v0 = vf and I don't see a justification for why this solution can be eliminated other than the fact that it would be a trivial solution where nothing at all happens to the plank. Thank you.

Whether the colliding ball bounces back or moves in the original direction depends on the ratio of masses. Remember the case of two colliding point masses. When one is in rest, the lighter incoming ball bounces back, one with equal mass stops and a heavier one moves in the original direction. The masses are equal here. in case m/M<1/4 the ball would bounce back.

You get v_f=vo a possible solution. You got v_f from the conservation equations and they are also valid when nothing happens: the ball does not touch the plank.

ehild

 

[WannabeNewton]

Whether the colliding ball bounces back or moves in the original direction depends on the ratio of masses. Remember the case of two colliding point masses. When one is in rest, the lighter incoming ball bounces back, one with equal mass stops and a heavier one moves in the original direction. The masses are equal here. in case m/M<1/4 the ball would bounce back.

You get v_f=vo a possible solution. You got v_f from the conservation equations and they are also valid when nothing happens: the ball does not touch the plank.

ehild

Thank you. So when I start I should not assume anything about the direction of the final velocity and that way its sign will come naturally out of what I choose for M,m?

 

[ehild]

Thank you. So when I start I should not assume anything about the direction of the final velocity and that way its sign will come naturally out of what I choose for M,m?

Yes. Decide what direction you choose as positive, and do not assume anything about the direction of the unknown velocity. It will come out.

ehild

 

[Nickie]

First of all, great job on using conservation of momentum and energy to solve this problem. Your approach is correct and your calculations seem to be accurate.

To address your concerns, it is important to note that the direction of the final velocity of the ball (v_{f}) depends on the relative masses of the ball and the plank. In the case where m = M, the final velocity of the ball will be in the opposite direction of its initial velocity (v_{0}). This is because the masses are equal, so the momentum of the ball needs to be conserved and the only way to do that is for the ball to bounce back with the same magnitude of momentum but in the opposite direction.

However, in the case where m < M, the final velocity of the ball will be in the same direction as its initial velocity. This is because the larger mass of the plank will have a greater influence on the final velocity, and the ball will continue in the same direction as before but with a smaller magnitude of velocity.

Regarding the case of v_{0} = v_{f}, this would indeed result in a trivial solution where nothing happens to the plank. This is because the ball would simply come to rest after the collision, and the plank would not experience any rotation or translation.

In summary, the direction of the final velocity (v_{f}) depends on the relative masses of the ball and the plank. In the case where m = M, the ball will bounce back and the plank will experience both rotation and translation. In the case where m < M, the ball will continue in the same direction but with a smaller velocity, and the plank will only experience rotation. The case of v_{0} = v_{f} would result in no change to the motion of the plank.