Question

Uma pequena esfera de massa m = 0,10 kg é abandonada de uma altura h = 5,0 m sobre o solo e volta a subir com 60% da velocidade de chegada ao chão. Adotando a aceleração da gravidade igual a 10m/s² e supondo desprezível a resistência do ar:
a) calcule a altura tingida pela esfera após quicar no chão.
b) calcule a energia consumida durante o seu choque com o solo.

Answer 1

$Ola'~~~~\mathbb{BRUNINHA} \\ \\ Ilustremos~para~entender: \\ \\ ~~~~~~~~~~~\textcircled{A} ----\bigotimes~~~~V_A=0~~~~Y_A=5m \\ ~~~~~~~~~~~~~~~~~~~~~~~~\bigg|~ \downarrow \\ ~~~~~~~~~~~~~~~~~~~~~~~~\bigg|\bigotimes~~~~~~\bigotimes---\textcircled{D} ~~Y_D=?~~~V_D=0\\ ~~~~~~~~~~~~~~~~~~~~~~~~\bigg|~\downarrow~~~~~~~\uparrow \\ ~~~~~~~~~~~\textcircled{B}---\bigg|~\bigotimes~~~~~\bigotimes ---\textcircled{C}\\ \overline{~V_B=?~~Y_B=0~~|~~~~~~~~~~~~V_{60\%=?~~~Y_C=0}}$

$\mathbb{ttttttttttttttttttttttttttttttttt} \\ Primeiro~calculemos~com~que'~velocidade~chega~ao~chao \\ temos: \\ \underbrace{Ec_A}_{0}+Ep_A=Ec_B+\underbrace{Ep_B}_{0} \\ \\ Ep_A=Ec_B \\ \\ \frac{1}{2}m.V\²=m.g.h~~--\ \textgreater \ cortamos~(m), fica \\ \\ \frac{1}{2}V\²=g.h~~--\ \textgreater \ temos[g=10m/s\²~~e~~h=5m~] \\ \\ \frac{1}{2}V\²=10.5 \\ \\ V= \sqrt{100} \\ \\ \boxed{V_B=10m/s } \\ \\ \mathbb{tttttttttttttttttttttttttttttttttt} \\ Respondendo~a pergunta~a)~temos: \\ 60\%V_B=V_C$

$\frac{60}{100}.10=V_C \\ \\ \boxed{V_C=6m/s} } \\ \\ Ec_C+\underbrace{Ep_C}_{0}=\underbrace{Ec_D}_{0}+Ep_D \\ \\ Ec_C=Ep_D \\ \\ \frac{1}{2}m.V_C\²= m.g.h_D~~--\ \textgreater \ cortamos~(m), temos \\ \\ \frac{1}{2}V_C\²=g.h_D~~--\ \textgreater \ temos~[V_C=6m/s~~e~g=10m/s\²] \\ \\ \frac{1}{2}.(6)\²=10.h_D~~--\ \textgreater \ isolando~[h_D] \\ \\ \boxed{\boxed{h_D=1,8m } } \\ \\ \mathbb{tttttttttttttttttttttttttttttttttt} \\ Respondendo~a ~pergunta~(b)~temos: \\ \\ \underbrace{Ec_A}_{0}+Ep_A=Ec_B+\underbrace{Ep_B}_{0}$

$Ep_A=Ec_B \\ \\ m.g.h=Ec_B \\ \\ 0,10.(10).(5)=Ec_B \\ \\ \boxed{\boxed{Ec_B=5J}}\\ \\ \\ \mathbb{iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii}\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Espero~ter~ajudado!!$