What is the largest speed this projectile can have without causing the cable to break?

Question

A 20.0 wood ball hangs from a 2.10 -long wire. The maximum tension the wire can withstand without breaking is 600 . A 0.900 projectile traveling horizontally hits and embeds itself in the wood ball.

Answer 1

[Edit -- I disagree with rhsaunders. His analysis assumes that the tension in the wire reaches its maximum when the ball reaches its maximum height in its swing--but in fact that is when the tension is at its _minimum_. The maximum tension is at the _bottom_ of the arc.]

Your numbers don't have any units attached; that makes a rather big difference to the answer. So I will generalize, and you can fill in the values yourself.

Variables:
M = mass of ball
m = mass of projectile
R = length of wire
Tmax = maximum tension the wire can withstand.
Vp = speed of projectile

Let V = the horizontal speed of the ball/projectile mass immediately after the projectile hits. By conservation of momentum:

(Vp)(m) = V(M+m)
or:
V = (Vp)(m) / (M+m)

Since the ball is constrained to swing in a circular arc, its centripetal acceleration (immediately after impact) is:

a_c = V² / R
= [(Vp)(m) / (M+m)]² / R

a_c is in an upward direction. So, by Newton's 2nd Law:

F_net_upward = (M+m)(a_c)

Furthermore, F_net_upward = (tension)–(weight) = T–(M+m)g. So:

T–(M+m)g = (M+m)(a_c)

Substituting our previous expression for a_c:

T–(M+m)g = (M+m) [(Vp)(m) / (M+m)]² / R

Next, just solve for Vp, then substitute Tmax for T.

Answer 2

Calculate the maximum angle (which will be just short of horizontal) at which 20.9 loading won't exceed 600, i.e. arctan(600/20.9). Assume that the ball will momentarily be stationary at that position, as motion would imply centripetal force, increasing the loading. Calculate the work done to lift the ball to that position, and from that the momentum imparted to the ball to make it happen. Divide this momentum by the projectile's mass to get its velocity.