Vận tốc trung bình của người đó trên quãng đường S là :

\(v_{tb}=\dfrac{s_1+s_2+s_3+...+s_n}{t_1+t_2+t_3+...+t_n}\)

Ta có : \(v_{tb}=\dfrac{v_1.t_1+v_2t_2+v_3t_3+...+v_nt_n}{t_1+t_2+t_3+...+t_n}=v_{min}.\dfrac{\dfrac{v_1}{v_{min}}.t_1+\dfrac{v_2}{v_{min}}.t_2+\dfrac{v_3}{v_{min}}.t_3+...+\dfrac{v_n}{v_{min}}.t_n}{t_1+t_2+t_3+...+t_n}\)

Do : \(\dfrac{v_1}{v_{min}};\dfrac{v_1}{v_{min}}...\dfrac{v_1}{v_{min}}>1nên\dfrac{v_1}{v_{min}}.t_1+\dfrac{v_1}{v_{min}}.t_2+\dfrac{v_1}{v_{min}}.t_3+...+\dfrac{v_1}{v_{min}}.t_n>t_1+t_2+t_3+...+t_n\)

=> vmin < vtb

Ta có :

\(v_{tb}=\dfrac{v_1t_1+v_2t_2+...+v_nt_n}{t_1+t_2+t_3+...+t_n}=v_{max}.\dfrac{\dfrac{v_1}{v_{max}}.t_1+\dfrac{v_2}{v_{max}}.t_2+...+\dfrac{v_n}{v_{max}}.t_n}{t_1+t_2+t_3+...+t_n}\)

Do : \(\dfrac{v_1}{v_{max}};\dfrac{v_1}{v_{max}}...\dfrac{v_1}{v_{max}}< 1\)

Nên : \(\dfrac{v_1}{v_{max}}.t_1+\dfrac{v_1}{v_{max}}t_2+\dfrac{v_1}{v_{max}}t_3+...+\dfrac{v_n}{v_{max}}.t_n< t_1+t_2+...t_n\)

=> vmax > vtb