Iniciante (Solução por João Guilherme Araújo)
Vamos adotar um sistema cartesiano "torto", onde o eixo x coincide com o plano inclinado e a origem é onde o corpo está antes de ser arremessado.
Assim, as equações para x e y são:
Vemos que ou
Substituindo em x:
Intermediário (Solução por Victor Sales)
Para nossa análise, podemos dizer que o carro da empresa sai de um ponto G ao mesmo tempo que o cientista pega o trem. Ou seja, se o trem leva um tempo
para ir deixar o Succa na casa de Sictor, a distância da casa deste até o ponto G será
, como na figura, onde
é a velocidade da limousine.
Na figura,
é a velocidade média do trem.
![V_T](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_836249d25ff1e3569fb77f337d1ed248.gif?w=640&ssl=1)
![ii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_a5efa8186bd4a083c7b71c83fc4135e2.gif?w=640&ssl=1)
![T = {\Delta t}_1 + {\Delta t}_2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_8e0bb8e17e8d8479bd8c77b360519afd.gif?w=640&ssl=1)
![T'](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_6f3357ae1d6de5c7df30cf8503177f87.gif?w=640&ssl=1)
![iii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_d63c15ce124eced5755c38457f207dac.gif?w=640&ssl=1)
![{{\Delta t}_1}' = {\Delta t}_1 - {\Delta t}_0](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_e4de3bcb805828a3da768b324653c77a.gif?w=640&ssl=1)
![{\Delta t}_0](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_e3dcf519eae326ed82aaa0d1645d34e0.gif?w=640&ssl=1)
![iv)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_2391b356adb67f210893125e4e1489ed.gif?w=640&ssl=1)
![S_c](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_a4dae485d3b9dff2b68583c536cce145.gif?w=640&ssl=1)
![x_c](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_37f14e7725ae8fec0cacf956a69e3f6e.gif?w=640&ssl=1)
![x = 0](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_3dad28281778d5ef4b7a78c7bc7a6b09.gif?w=640&ssl=1)
![v)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_5fa72f546a9fbca5a519c2d848b916b6.gif?w=640&ssl=1)
![V](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_5206560a306a2e085a437fd258eb57ce.gif?w=640&ssl=1)
![{{\Delta t}_2}'](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_775cd46fc483c54cce90b818546883d8.gif?w=640&ssl=1)
![0](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_cfcd208495d565ef66e7dff9f98764da.gif?w=640&ssl=1)
![vi)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_511428aaceaf9f8fd01cae0f66eabda1.gif?w=640&ssl=1)
![vii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_8e4652b71faaafd62e347a90cb04d80a.gif?w=640&ssl=1)
![{{\Delta t}_3}'](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_c9e6e6a59af04e8a55000a47a5007a13.gif?w=640&ssl=1)
![Viii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_1272d05565546573bb5bef79651e9e48.gif?w=640&ssl=1)
![T' = {{\Delta t}_1}' + {{\Delta t}_2}' + {{\Delta t}_3}'](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_a9cff1b2cb4a53bbcc3dc53e31cf1746.gif?w=640&ssl=1)
Substituindo
e
, temos:
![\eta = 11](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_dd20738560e91856c10cde947b98d891.gif?w=640&ssl=1)
![{\Delta t}_0 = 1 h](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_b9820239a96aeccadad913632c03c762.gif?w=640&ssl=1)
Avançado (Solução por Victor Sales)
![i)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_71e30cee73fa1933d3cd6959ef542ab9.gif?w=640&ssl=1)
![V_{\alpha} = V_{n} = V](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_9aa36c4d9388327fbdb9b031f1df4f84.gif?w=640&ssl=1)
![ii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_a5efa8186bd4a083c7b71c83fc4135e2.gif?w=640&ssl=1)
![iii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_d63c15ce124eced5755c38457f207dac.gif?w=640&ssl=1)
![E_{He} = \gamma m_{\alpha} c^2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_2f306c339924ab884d01d458b6c96679.gif?w=640&ssl=1)
![E_n = \gamma m_{n} c^2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_aa951edc34fb8475527b66ff0c7a1d08.gif?w=640&ssl=1)
![\gamma = \frac{1}{\sqrt{1 - (\frac{v}{c})^2}}](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_c20ee625b1d5353fd9090e7c85456684.gif?w=640&ssl=1)
![iv)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_2391b356adb67f210893125e4e1489ed.gif?w=640&ssl=1)
![m_p = m_n = m](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_93ac369e69b5f21c437243adce38c210.gif?w=640&ssl=1)
![E_{\alpha} = 4 m \gamma c^2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_46f4bca7d5f4cb4b644ee7f92fa45322.gif?w=640&ssl=1)
![E_n = m \gamma c^2](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_5e95dccd6265574935605c73d9fdf9d7.gif?w=640&ssl=1)
Substituindo em
:
![(ii)](https://i0.wp.com/noic.com.br/wp-content/plugins/latex/cache/tex_0aff6a0421e79365ab8fcc70d0385828.gif?w=640&ssl=1)
Ou seja: