Savchenko Problem 1.2.16

The key insight is that identical graph shapes with different time scales reveal how velocity and acceleration scale with time. Since both particles trace the same curve on their respective graphs, their position functions have the same mathematical form—but time is compressed or stretched by a factor of 4 between them. When you change the time scale, velocities scale inversely with time (faster clock = steeper slope per unit real time), while accelerations scale as the inverse square of the time scale. Use the chain rule: if particle 1's time axis is stretched 4× relative to particle 2's, then $v_1/v_2 = 1/4$ and $a_1/a_2 = 1/16$.

The key insight is that identical graph shapes with different time scales reveal how velocity and acceleration scale with time. Since both particles trace the same curve on their respective graphs, their position functions have the same mathematical form—but time is compressed or stretched by a factor of 4 between them. When you change the time scale, velocities scale inversely with time (faster clock = steeper slope per unit real time), while accelerations scale as the inverse square of the time scale. Use the chain rule: if particle 1's time axis is stretched 4× relative to particle 2's, then $v_1/v_2 = 1/4$ and $a_1/a_2 = 1/16$.