Hello, unifieda!

This is a strange one . . . an "eyeball" problem

Just *look* at it . . .nothing fancy is needed!

A stone is thrown from a rooftop at time $\displaystyle t = 0.$

Its position at time $\displaystyle t$ is given by: .$\displaystyle r(t) \:=\:(10t,\: -5t,\: 6.4-4.9t^2)$

The origin is at the base of the building, which is on flat ground.

Distance is measured in meters and time in seconds.

(a) How high is the rooftop?

I know that the initial velocity r'(t) = 0 at t = 0. . This is not true! We are given three functions: . $\displaystyle \begin{Bmatrix}x \;=\;10t \\ y \;=\;-5t \\ z \;=\;6.4-4.9t^2 \end{Bmatrix}$

I would *guess* the following:

$\displaystyle x$ is the east-west location of the stone.

. . It is moving east at 10 m/sec. **

$\displaystyle y$ is the north-south location of the stone.

. . It is moving south at 5 m/sec. **

$\displaystyle z$ is the height of the stone above the ground.

At $\displaystyle t = 0,\;z = 6.4$

The stone began at 6.4 m above the ground.

. . Therefore, the height of the building is 6.4 meters.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

**

The stone is thrown horizontally from the building.

Looking down, the scene looks like this: Code:

10t
A * - - - - - *
* θ |
* | 5t
* |
* B

The stone is thrown from $\displaystyle A$ in the direction of $\displaystyle B.$

Its speed is: .$\displaystyle \sqrt{10^2 + 5^2} \:=\:\sqrt{125} \:\approx\:11.2$ m/sec.

Its direction is: .$\displaystyle \theta \:\approx\;26.7^o$ south of east.

So, you see, the initial velocity was not zero.