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Recent experiments have discovered two exponents in the $pp$ elastic differential cross sections with two different slope parameters, of the order $(16-20)$~GeV$^{-2}$ and $(4-4.8)$~GeV$^{-2}$ in the regions $ -t \la 0.5 $~GeV$^2$ and $ -t \ga 1$~GeV$^2$, respectively. We suggest a simple model of the $pp$ elastic scattering with two types of particle exchanges: 1) when the exchanged particle transfers the momentum $\veQ$ from a quark of the proton $p_1$ to one quark in another proton $p_2$, producing the slope $B_1$; 2) when the transfer occurs from two quarks in the $p_1$ to two quarks in the $p_2$, giving the exponent with the slope $B_2$. The resulting amplitude is proportional to the product of the form factors of two protons, depending on $\veQ$, but with different coefficients in the cases 1) and 2). Using the only parameter - the proton charge radius $r^2_{ch}= 0.93$~fm$^2$, one obtains $B_1 = 16$~GeV$^{-2}$,~$ B_2 = 4$~GeV$^{-2}$ with the strict value of the ratio, $\frac{B_1}{B_2} = 4.0$, independent of $r_{ch}$. These predictions are surprisingly close to the data both in the $pp$ and in the $\bar p p$ differential cross sections. Comparison to experimental data and theoretical approaches is discussed, together with possible implications for the future development of the theory.