# How can I find the angle of deviation from the floor when a bullet strikes a block?

#### Chemist116

The problem is as follows:

The diagram from bellow shows a bullet which strikes a block hanging from a ceiling. The mass of the block is $20\,g$. Such bullet impacts to the block of mass of $980\,g$ and gets embedded on it. Find the angle that deflects the maximum from the vertical. You may use $g= 10 \,\frac{m}{s}$

The alternatives given are:

$\begin{array}{ll} 1.&20^{\circ}\\ 2.&30^{\circ}\\ 3.&37^{\circ}\\ 4.&53^{\circ}\\ 5.&60^{\circ}\\ \end{array}$

I'm stuck on this problem. What I attempted to do was to use the principle of conservation of momentum for an inelastic collision as follows:

$p_i=p_f$

Therefore

$mv_1=\left(m_1+m_2\right)v_2$

$20\times 50 = (20+980)v_f$

$v_f=1\,\frac{m}{s}$

But I'm stuck on how to find the angle can someone help me with this please?.

#### topsquark

Math Team
Once the bullet hits the block then it's a conservation of energy problem... It's a pendulum. How do you calculate how high the pendulum will go if we give the mass a push? ($$\displaystyle P_f + KE_f = P_i + KE_i$$.)

By the way, this apparatus is called a "ballistic pendulum" for that reason.

-Dan

#### Chemist116

Once the bullet hits the block then it's a conservation of energy problem... It's a pendulum. How do you calculate how high the pendulum will go if we give the mass a push? ($$\displaystyle P_f + KE_f = P_i + KE_i$$.)

By the way, this apparatus is called a "ballistic pendulum" for that reason.

-Dan
I don't understand the notation of $P_f$ what's exactly meaning?. Is it the momentum?. I'm still stuck on what should be done to solve this problem.

If it deviates from the vertical then the block will raise a certain height. This height will be I'm assuming

$R-R\cos\phi$

But then what... I'm stuck exactly how should I use this information?. Perhaps relate it with the centripetal force and the weight?.

I still can't do it alone. Can you help me with this?.

#### skeeter

Math Team
topsquark is using $P$ to indicate gravitational POTENTIAL energy ... maybe he should have used $U_g$ instead to avoid confusing you.

setting $h_0=0$, the initial (post collision) energy is all kinetic; use conservation of momentum to determine the speed of the bullet/block combination after the collision

at max height of the pendulum, all the energy is gravitational potential

$\dfrac{1}{2}(m+M)v_f^2 = (m+M)gh_{max} \implies h_{max} = \dfrac{v_f^2}{2g}$ (mind the units of $h_{max}$)

$\cos{\theta} = \dfrac{0.10 - h_{max}}{0.10}$

side note ... are you receiving any formal instruction (including the requisite labs) for all these mechanics problems you are posting?

topsquark