Question

For an object falling freely from rest, show that the distance traveled during each successive second increases in the ratio of successive odd integers (1, 3, 5, etc.). (This was first shown by Galileo.) See Figs. 2–19 and 2–22.

Short answer

It is proved that the distance traveled during each successive second increases in the ratio of successive odd integers.

Step-by-step solution

Step 1. Free fall of an object

Free fall is represented as the occurrence of a specific object being released from a height toward the ground.

For example, meteors falling toward the ground, movement of spacecraft in continuous orbit, and many others.

The relation from the equation of motion is given by.

$S=ut+\frac{1}{2}g{t}^2$

Here, u is the initial speed whose value is zero, g is the gravitational acceleration, S is the distance traveled, and t is the time.

On plugging the values in the above relation, you get.

role="math" localid="1643005824541" $\begin{array}{c}S\left(t\right)=0\times t+\frac{1}{2}g{t}^2\\ S\left(t\right)=\frac{1}{2}g{t}^2\dots \left(i\right)\end{array}$

Step 2. Calculation of distance traveled

Substitute $t+1$for $t$in equation (i).

$S\left(t+1\right)=\frac{1}{2}g{\left(t+1\right)}^2$

The relation to show the distance traveled during each successive second is given by.

role="math" localid="1643005934633" $\begin{array}{l}S\left(t+1\right)-S\left(t\right)=\left[\frac{1}{2}g{\left(t+1\right)}^2-\frac{1}{2}g{t}^2\right]\\ S\left(t+1\right)-S\left(t\right)=\frac{g}{2}\left[2t+1\right]\dots \left(ii\right)\end{array}$

At time $t=0$, the value of time $2t+1$is 1.

At time $t=1$, the value of time $2t+1$is 3.

At time $t=2$, the value of time $2t+1$is 5.

Thus, it is shown that the distance traveled during each successive second increases in the ratio of successive odd integers.