Falling ball — velocity and acceleration

Problem

Difference between velocity and acceleration. A ball falls from rest. After $1\;\text{s}$ its velocity is $\sim 10\;\text{m/s}$, after $2\;\text{s}$ $\sim 20\;\text{m/s}$. Explain in five lines in what sense “the velocity changes” and what the acceleration represents.

Solution

The velocity tells us how fast the ball is moving at a given instant: after $1\;\text{s}$ it is $10\;\text{m/s}$, after $2\;\text{s}$ it is $20\;\text{m/s}$. The fact that this number differs from one instant to the next means that “the velocity changes”.

The acceleration measures precisely the rate at which the velocity changes, that is, the ratio between the change in velocity and the time taken to produce it:

$a = \frac{\Delta v}{\Delta t} = \frac{20 - 10}{2 - 1} = 10\;\mathrm{m/s^2}$

Every second the velocity increases by $10\;\text{m/s}$: this constant value is the acceleration due to gravity $g$. In the graph $v(t)$ is a straight line starting from the origin, and its (constant) slope is precisely $a = g$.

$\ev{a = \frac{\Delta v}{\Delta t} = 10\;\mathrm{m/s^2} = g}$

Collegamenti

Argomenti: Kinematics Concetti: Free fall · Acceleration