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

Jun 2017
302
6
Lima, Peru
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
May 2013
2,385
996
The Astral plane
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
 
Jun 2017
302
6
Lima, Peru
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
Jul 2011
3,212
1,734
Texas
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?
 
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