Conservation of momentum and wood ball problem

[Amria]

Problem:
A 20.0 kg wood ball hangs from a 1.50 m-long wire. The maximum tension the wire can withstand without breaking is 300 N. A 0.900 kg projectile traveling horizontally hits and embeds itself in the wood ball.
What is the largest speed this projectile can have without causing the cable to break?

What I have tried:

I tried to use the tension formula (T = mv^2/r) along with the conservation of momentum formula.
First I tried:
v = sqrt(Tr/m) = sqrt(300*1.5/20) = 4.74
and then m1v1 = m2v2 = mfvf
so .9v = 20.9 *4.74 = 110.2 m/s which was wrong

then I tried:
v = sqrt(Tr/m) = sqrt(300*1.5/20.9) = 4.64
so .9v = 20.9 * 4.64 = 107.8, which is also wrong

Can anyone point out where I'm making a mistake? I think the theory is sound, but I can't figure out where I'm going wrong.

 

[CaptainZappo]

The NET force is F=(mv^2)/r. The two forces acting on the hanging mass(es) are gravity and the tension in the wire. I cannot see that you've accounted for the former. Go back, draw a free-body diagram with the gravitational force included, and see where your thought leads you.

Problem:
A 20.0 kg wood ball hangs from a 1.50 m-long wire. The maximum tension the wire can withstand without breaking is 300 N. A 0.900 kg projectile traveling horizontally hits and embeds itself in the wood ball.
What is the largest speed this projectile can have without causing the cable to break?

What I have tried:

I tried to use the tension formula (T = mv^2/r) along with the conservation of momentum formula.
First I tried:
v = sqrt(Tr/m) = sqrt(300*1.5/20) = 4.74
and then m1v1 = m2v2 = mfvf
so .9v = 20.9 *4.74 = 110.2 m/s which was wrong

then I tried:
v = sqrt(Tr/m) = sqrt(300*1.5/20.9) = 4.64
so .9v = 20.9 * 4.64 = 107.8, which is also wrong

Can anyone point out where I'm making a mistake? I think the theory is sound, but I can't figure out where I'm going wrong.

 

[m2003]

Your approach using the tension formula and conservation of momentum is correct. However, the mistake you are making is in the calculation of the final velocity of the system. The correct equation for conservation of momentum is m1v1 + m2v2 = (m1+m2)vf. In your calculation, you are using mfvf instead of (m1+m2)vf, which is incorrect. The correct calculation would be:

v = sqrt(Tr/m) = sqrt(300*1.5/20.9) = 4.64
so (0.9+20.9)vf = 20.9 * 4.64 = 107.8
vf = 107.8/21.8 = 4.95 m/s

Therefore, the largest speed the projectile can have without causing the cable to break is 4.95 m/s.