Question

The auto in the sketch moves forward as the brakes are applied. A bystander says that during the interval of braking, the auto's velocity and acceleration are in opposite directions. Do you agree or disagree?

Answer 1

Answer:

The statement is true: velocity and acceleration have opposite directions in the interval of braking.

Explanation:

Let's say we have a velocity $v>0$.

The acceleration $a$ is the rate of change of the velocity $v$. This means that if $v$ is increasing during time, then $a$ must be positive. But if $v$ is decreasing over time, then $a$ will be negative (even though the velocity is positive).

Mathematically:

$a=\frac{dv}{dt}$

$v$ decreases ⇒$\frac{dv}{dt}$

⇒$a$.

Example:

$v(t)=e^{-t}>0 \\\\\frac{dv}{dt}=-te^{-t}$