Box falling from the shelf

Problem

A supermarket shelf is $1{,}5\;\text{m}$ above the ground. A customer accidentally drops a box of pasta. How much time does he have (at most) to try to catch it before it hits the ground? How much distance will his hand cover if it moves down with an average velocity of $1\;\text{m/s}$? Is it enough?

Solution

The box starts from rest and falls freely from a height $h = 1{,}5\;\text{m}$. The relationship between height and fall time is that of uniformly accelerated motion with acceleration $g$:

$h = \frac{1}{2}\,g\,t^2 \quad\Longrightarrow\quad t = \sqrt{\frac{2h}{g}}$

Substituting the data, with $g \approx 9{,}81\;\mathrm{m/s^2}$:

$t = \sqrt{\frac{2 \cdot 1{,}5}{9{,}81}} = \sqrt{0{,}306}\;\mathrm{s} \approx 0{,}55\;\mathrm{s}$

This is the maximum time available to intervene. If the hand moves down with average velocity $\bar v = 1\;\text{m/s}$, in the same interval it covers:

$d = \bar v \cdot t = 1 \cdot 0{,}55 = 0{,}55\;\mathrm{m}$

The hand only manages to move down about half a metre, but the box must travel $1{,}5\;\text{m}$: the attempt is therefore not enough to intercept it.

$\ev{t \approx 0{,}55\;\mathrm{s}, \quad d \approx 0{,}55\;\mathrm{m} < 1{,}5\;\mathrm{m}}$

Collegamenti

Argomenti: Kinematics Concetti: Free fall