Question

A 20 kg wood ball hangs from a 2.0-m-long wire. The maximum tension the wire can withstand without breaking is 400 N. A 1.0 kg projectile traveling horizontally hits and embeds itself in the wood ball. What is the greatest speed this projectile can have without causing the wire to break?

Short answer

The greatest speed that projectile can have without causing the wire to break is 90.3 ms^-1

Step-by-step solution

Step 1. Given information isMass of the ball, mb =  20 kgLength of the wire, l = 2 mMass of the projectile, mp = 1 kgMaximum tension that the wire can withstand, T = 400 N  

We need to find out the the greatest speed that this projectile can have without causing the wire to break.

Step 2. Explanation of the projectile with ball embedded in it after collision.

v_pi and v_f denotes the initial velocity of the projectile and the final velocity of the ball embedded with the projectile respectively.

After collision, the projectile with ball moves in a circular arc.

Step 3. Using the principle of conservation of momentum for the case of before and after collision to find the relation between initial and final velocity. 

Using the Principle of conservation of momentum,

m_pv_pi = (m_p + m_b)v_f

$$\begin{gathered} v_{pi}=\frac{(m_p+m_b)}{m_p}{v}_f \\ {v}_{pi}=\frac{(1+20)}{1}{v}_f \\ {v}_{pi}=21v_f \end{gathered}$$

To find v_pi , the value of v_f is required

After collision, the ball with projectile moves in a circular arc.

The force acting along wire is given as

$$\begin{gathered} F=T-(m_P+m_b)g\cos \theta \\ As\theta isverysmall \\ F\approx T-(m_P+m_b)g \end{gathered}$$

Centripetal force is also acting due to the wire. Therefore,

$$\begin{gathered} F=\frac{(m_b+m_p){v_f}^2}{l} \\ Onequating, \\ T-(m_b+m_p)g=\frac{(m_b+m_p){v_f}^2}{l} \\ {v}_f=\sqrt{\frac{[T-(m_b+m_p\left)g\right]l}{(m_b+m_p)}}=\sqrt{\frac{[400-(1+20)\times 9.8]\times 2}{(1+20)}}=4.3m{s}^{-1} \end{gathered}$$

And it is calculated that

v_pf = 21v_f

Therefore,

v_pf= $21\times 4.3$

v_pf= 90.3 ms^-1

Hence, The greatest speed that projectile can have without causing the wire to break is 90.3 ms^-1