Drop stone off cliff - estimate height with time

[heartshapes]

Amazingly enough, you and your friends have climbed a very tall mountain and are standing right on the brink looking down. Someone in the group wonders how many feet is it down to the ground. You have the bright idea of using your Calculus to estimate the distance. You arrange someone to drop a stone and someone else to use her fancy wristwatch acting as a stopwatch and time the interval it takes for the tone to hit the ground. It takes 6.2 seconds for the stone to reach the ground. You make some quick calculations using your cellphone calculator and estimate the mountains height.

a) Explain using nice notations, diagrams, algebra, etc, how you determined the height of the mountain.

b) Now compute the height.

c) Estimate the velocity of the rock when it hits the ground.

Ok. So I know the the derivative of distance is velocity and the derivative of velocity is acceleration. But other then that I don't know where to start. I also know that distance = velocity * time . So once I find the distance I can do that to do c.

 

[galactus]

I have actually done this for fun.

We need \(\displaystyle 16t^{2}\) for the height.

We know that t=6.2

As for the average velocity:

\(\displaystyle v_{ave}=\frac{16(6.2)^{2}-16(0)^{2}}{6.2-0}\)

 

[mmm4444bot]

... I have actually done this for fun ...

Me, too!

(Although, I dropped my calculus book.)

Amazingly enough, you and your friends have climbed a very tall mountain ...

This seems somewhat pejorative (unless you're disabled).

... Ok. So I know the the derivative of distance is velocity ...

Actually, the derivative of displacement is velocity.

... I also know that distance = velocity * time . So once I find the distance I can do that to do c.

You could, but you'd be wrong. The velocity is CHANGING over time. That's the meaning of acceleration.

Elapsed Time / Velocity
1 second / -32 feet per second
2 seconds / -64 feet per second
3 seconds / -96 feet per second
4 seconds / -128 feet per second
5 seconds / -160 feet per second
6 seconds / -192 feet per second

Since you understand that velocity is the derivative of displacement, and you have the displacement (courtesy of Galactus), use calculus to find the velocity.

Cheers,

~ Mark

 

[heartshapes]

ok thank you both!