Elastic Collision between projectile and a pendulum, what's the initial speed?

[Fromaginator]

Homework Statement

A 29.0 g ball is fired horizontally with initial speed v₀ toward a 100 g ball that is hanging motionless from a 1.10 m-long string. The balls undergo a head-on, perfectly elastic collision, after which the 100 g ball swings out to a maximum angle θ_max = 50.0°
see image for question

Homework Equations

K_i + U_i = K_f + U_f
vp = √[2*g*L*(1 - cosθ)]
mb*v0 = mp*vp + mb*vb

The Attempt at a Solution

I suspect you have to use the mass ratio and it's relation to the ratio of momentum or kinetic energy, but I have been unsuccessful in getting the correct answer. The tried answers are availible along with the question in the image.

 

[Delphi51]

I got a different answer. But I can't just tell you - that would be spoiling your learning. If you show your calculation, I can suggest improvements.

 

[Fromaginator]

I think the problem with my calculations was that I used a mass ratio which I think only applies to inelastic but here's what I did
0.5*mp*vp² = mp*g*L*(1 - cosθ)

vp = √[2*g*L*(1 - cosθ)]

0.5*mb*v0² = 0.5*mb*vb² + 0.5*mp*vp²
mb*v0² = mb*vb² + mp*vp²
mb*vb² = mb*v0² - mp*vp²
vb = √[v0² - (mp/mb)*vp²]

v0 = √[v0² - (mp/mb)*vp²] + (mb/mp)*vp


mb = 27 g
mp = 100 g
vp = √[2*g*L*(1 - cosθ)]; L = 1.1 m, θ = 50º

vp = 2.775 m/s

mp/mb = 3.70

v0 = √[v0² - (3.7)*7.701] + (3.7)*2.775
v0 = √[v0² - 28.5] + 10.27
v0 - 10.27 = √[v0² - 28.5]
v0² - 20.54*v0 + 105.4 = v0² - 28.5
- 20.54*v0 + 105.4 = - 28.5

v0 = 6.52 m/s

I'm pretty sure the vp(final velocity of the pendulum is right)

 

[ideasrule]

v0 = √[v0² - (mp/mb)*vp²] + (mb/mp)*vp

This came from the conservation of momentum equation, but you made a mistake here. If you write out the full momentum equation, you'll see what it should be.

 

[Fromaginator]

I got it!
vp=[2mb/(mb+mp)]*vb
0.5*mvbf^2+mghf=0.5*mvbi^2+mghi
gL(1-costheta)=).5(Vo^2)(3364/16641)
Vo=6.17m/s

although when I actually did it there were a few more intermeadiate steps but I didn't feel it necessary to type out the algebra since square roots and fractions are hard to type.