Source: High school physics (Chinese)
Problem
An object starts from rest ($v_0 = 0$) and undergoes uniform acceleration $a$. Let $s_n$ be the total displacement in the first $n$ seconds (i.e., in a total time of $t=n$ seconds). Let $s_N$ (using Roman numerals in the original text) be the displacement during the $N$-th second (i.e., in the time interval from $t=N-1$ to $t=N$).
- Find the ratio of total displacements after consecutive integer seconds, $s_1 : s_2 : s_3 : \dots$.
- Find the ratio of displacements during consecutive one-second intervals, $s_I : s_{II} : s_{III} : \dots$.
- What is the physical meaning of the differences between displacements in consecutive seconds, such as $s_{II} - s_I$ and $s_{III} - s_{II}$? What is the relationship between these differences?
[Q1] The ratio of total displacements is the ratio of the squares of integers:
$$s_1 : s_2 : s_3 : \dots = 1^2 : 2^2 : 3^2 : \dots = 1 : 4 : 9 : \dots$$[Q2] The ratio of displacements during consecutive one-second intervals is the ratio of consecutive odd integers (Galileo's law of odd numbers):
$$s_I : s_{II} : s_{III} : \dots = 1 : 3 : 5 : \dots$$[Q3] The difference in displacement between any two consecutive one-second intervals is constant and equal to the magnitude of the acceleration, $a$.
$$s_{N+1} - s_N = a$$This represents the constant increase in distance covered per unit time, which is a direct result of uniform acceleration.
- The motion of an object starting from rest ($v_0 = 0$) with uniform acceleration $a$ is described by the kinematic equation for displacement $s$: $$s(t) = v_0 t + \frac{1}{2}at^2 = \frac{1}{2}at^2$$ [Q1] Find the ratio of total displacements after consecutive integer seconds, $s_1 : s_2 : s_3 : \dots$. The total displacement $s_n$ after $n$ seconds is found by setting $t=n$: $$s_n = s(t=n) = \frac{1}{2}an^2$$ The displacements after $n=1, 2, 3, \dots$ seconds are: $s_1 = \frac{1}{2}a(1)^2 = \frac{1}{2}a$ $s_2 = \frac{1}{2}a(2)^2 = 4 \left(\frac{1}{2}a\right)$ $s_3 = \frac{1}{2}a(3)^2 = 9 \left(\frac{1}{2}a\right)$ The ratio is found by taking the ratio of these values: $$s_1 : s_2 : s_3 : \dots = \frac{1}{2}a(1^2) : \frac{1}{2}a(2^2) : \frac{1}{2}a(3^2) : \dots$$ $$s_1 : s_2 : s_3 : \dots = 1^2 : 2^2 : 3^2 : \dots = 1 : 4 : 9 : \dots$$ [Q2] Find the ratio of displacements during consecutive one-second intervals, $s_I : s_{II} : s_{III} : \dots$. The displacement during the $N$-th second, $s_N$, is the difference between the total displacement at time $t=N$ and the total displacement at time $t=N-1$. $$s_N = s(t=N) - s(t=N-1)$$ $$s_N = \frac{1}{2}aN^2 - \frac{1}{2}a(N-1)^2 = \frac{1}{2}a [N^2 - (N^2 - 2N + 1)] = \frac{1}{2}a(2N-1)$$ The displacements during the 1st ($N=1$), 2nd ($N=2$), 3rd ($N=3$), etc., seconds are: $s_I = \frac{1}{2}a(2(1)-1) = \frac{1}{2}a(1)$ $s_{II} = \frac{1}{2}a(2(2)-1) = \frac{1}{2}a(3)$ $s_{III} = \frac{1}{2}a(2(3)-1) = \frac{1}{2}a(5)$ The ratio is therefore: $$s_I : s_{II} : s_{III} : \dots = \frac{1}{2}a(1) : \frac{1}{2}a(3) : \frac{1}{2}a(5) : \dots$$ $$s_I : s_{II} : s_{III} : \dots = 1 : 3 : 5 : \dots$$ [Q3] What is the physical meaning of the differences between displacements in consecutive seconds, such as $s_{II} - s_I$ and $s_{III} - s_{II}$?
- What is the relationship between these differences? Let's find the general expression for the difference between displacements in consecutive seconds, $s_{N+1} - s_N$: $$s_{N+1} - s_N = \frac{1}{2}a(2(N+1)-1) - \frac{1}{2}a(2N-1)$$ $$s_{N+1} - s_N = \frac{1}{2}a[(2N+2-1) - (2N-1)] = \frac{1}{2}a(2N+1 - 2N+1) = \frac{1}{2}a(2)$$ $$s_{N+1} - s_N = a$$ This shows that the difference is constant and equal to the acceleration $a$. **Physical Meaning:** The velocity of the object increases linearly with time ($v=at$). This means the object travels a greater distance in each successive time interval. The difference $s_{N+1} - s_N$ represents the *increase* in distance traveled in a one-second interval compared to the previous one-second interval. This constant increase in distance covered per second is a direct consequence of the constant rate of increase of velocity (i.e., constant acceleration). **Relationship:** The differences are all equal to each other and equal to the magnitude of the uniform acceleration, $a$. $$s_{II} - s_I = s_{III} - s_{II} = \dots = a$$