If an object with mass m is dropped from rest, one model for its speed v after t seconds, taking air resistance into account, is

v $=$

mgc

$(1$ e$$ct$$m$),$

where g is the acceleration due to gravity and c is a positive constant. $($Note that the air resistance is proportional to the velocity of the object; c is the proportionality constant.$)$

$($a$)$

Calculate

lim$$t$$$$v$.$

$0$

What is the meaning of this limit in the context of this problem?

It is the speed the object approaches as time goes on$.$It is the time it takes for the object to stop.$$It is the time it takes the object to reach its maximum speed.It is the speed the object reaches before it starts to slow down.

$($b$)$

For fixed t$,$ use l'Hospital's Rule to calculate

lim$$c$0+$v$.$

$0$

What can you conclude about the velocity of a falling object in a vacuum?

An object falling in a vacuum will accelerate at a slower rate than an object not in a vacuum.The heavier the object is the faster it will fall in a vacuum.$$The velocity of a falling object is proportional to its mass in a vacuum.The velocity of a falling object in a vacuum is directly proportional to the amount of time it falls.