Question

12. Um automóvel faz metade de um percurso com velocidade média de 40 km/h e a
outra metade com velocidade média de 60 km/h. Qual a velocidade média do
automóvel no percurso todo?
a) 16 km/h
d) 24 km/h
b) 48 km/h
e) 32 km/h
c) 64 km/h​

Answer 1

Alternativa B) 48 km/h

Answer 2

$\mathrm{Seja}\ A\ \mathrm{a\ 1^a\ parte\ do\ percurso\ e}\ B\ \mathrm{a\ 2^a\ parte.}$

$\mathrm{Logo},\ v_A=40\ \mathrm{km/h}\ \mathrm{e}\ v_B=60\ \mathrm{km/h}.\ \mathrm{Al\acute{e}m\ disso,}\ d_A=d_B=x.$

$\mathrm{O\ tempo\ gasto\ em}\ A\ \mathrm{ser\acute{a}}\text{:}$

$v_A=\dfrac{d_A}{t_A}\Longrightarrow t_A=\dfrac{d_A}{v_A}\Longrightarrow t_A=\dfrac{x}{40}$

$\mathrm{O\ tempo\ gasto\ em}\ B\ \mathrm{ser\acute{a}}\text{:}$

$v_B=\dfrac{d_B}{t_B}\Longrightarrow t_B=\dfrac{d_B}{v_B}\Longrightarrow t_A=\dfrac{x}{60}$

$\mathrm{O\ tempo\ total\ ser\acute{a}}\text{:}$

$t_{\mathrm{total}}=t_A+t_B=\dfrac{x}{40}+\dfrac{x}{60}=\dfrac{60x+40x}{(60)(40)}\Longrightarrow t_{\mathrm{total}}=\dfrac{x}{24}$

$\mathrm{A\ dist\hat{a}ncia\ total\ ser\acute{a}}\text{:}$

$d_{\mathrm{total}}=x+x\Longrightarrow d_{\mathrm{total}}=2x$

$\mathrm{A\ velocidade\ m\acute{e}dia\ do\ percurso\ todo\ ser\acute{a}}\text{:}$

$v_m=\dfrac{d_{\mathrm{total}}}{t_{\mathrm{total}}}=\dfrac{2x}{\frac{x}{24}}=2x\bigg(\dfrac{24}{x}\bigg)\ \therefore\ \boxed{v_m=48\ \mathrm{km/h}}$

$\mathbf{Alternativa\ B.}$