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

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

Step-by-step 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=(dv)/(dt)`

`v` decreases ⇒
`(dv)/(dt)<0`

⇒
`a<0`.

Example:

`v(t)=e^(-t)>0 \\\\(dv)/(dt)=-te^(-t) <0\\\\a<0.`