Dynamical Systems: Modeling of ball dropped from 100 feet

[moy1989]

Hey everyone, I'm trying to review some math and I'm having difficulty with the following problem:

Suppose you drop a ball from the top of a building that is 100 feet tall.

(a) Construct a mathematical model to estimate how long it takes the ball to reach the ground. HINT: An object falling near the surface of the earth in the absence of air friction accelerates downward at the rate of g = 32.2 ft/sec^2.

(b) Use calculus to solve the model and answer the question.

(c) After solving the model for your estimate, suppose you actually drop the ball and discover it takes 2.6 seconds to reach the ground. What do you conclude from this result?

This is what i have done so far:

(a) dy/dt = gy, g = 32.2 ft/sec^2

dy/dt = 32.2y, y(0) = 100

(b) Using calculus I found

y = 100e^(-32t) satisfies the equation above

verification:

if y(t) = 100e^(-32t)
y'(t) = -32(100)e^(-32t)
y'(t) = -32 * y(t)
At t = 0, y(0) = 100e^0 = 100

I don't know how to find the time it takes the ball to hit the ground; I tried the following:

y = 100e^(-32t)

ln(y) = -3200t
t = ln(y)/-3200

Am I doing this right?

Thanks for any help.

 

[DR._Glockman]

Re: Dynamical Systems: Modeling

Distance Formula: S(t) = -16t^2+Vo(t)+So, Vo = initial velocity, So = initial height, hence when the ball hits the ground

S(t) = (position function) = 0 = -16t^2+100 (initial velocity = 0, and initial height = 100, t= 2.5 sec.

If t = 2.6 sec, then 0 = (g/2)(2.6)^2+100, g/2 = -14.7929, g = -29.5858, hence air resistance counteracted the free falling object.

In other words instead of falling -32 feet per second per second, air resistance cause the object to fall -29.5858 feet per second per second, thereby hitting the ground in 2.6 seconds instead of 2.5 seconds.

 

[skeeter]

Re: Dynamical Systems: Modeling

Hey everyone, I'm trying to review some math and I'm having difficulty with the following problem:

Suppose you drop a ball from the top of a building that is 100 feet tall.

(a) Construct a mathematical model to estimate how long it takes the ball to reach the ground. HINT: An object falling near the surface of the earth in the absence of air friction accelerates downward at the rate of g = 32.2 ft/sec^2.

(b) Use calculus to solve the model and answer the question.

(c) After solving the model for your estimate, suppose you actually drop the ball and discover it takes 2.6 seconds to reach the ground. What do you conclude from this result?

This is what i have done so far:

(a) dy/dt = gy, g = 32.2 ft/sec^2

acceleration is the second time derivative of position, so start with d[sup:2g6ty82q]2[/sup:2g6ty82q]y/dt[sup:2g6ty82q]2[/sup:2g6ty82q] = -g ...
integrating w/r to time yields the velocity function, dy/dt = v = -gt + v[sub:2g6ty82q]0[/sub:2g6ty82q] ...
integrating once again yields the position function, y = -(1/2)gt[sup:2g6ty82q]2[/sup:2g6ty82q] + v[sub:2g6ty82q]0[/sub:2g6ty82q]t + y[sub:2g6ty82q]0[/sub:2g6ty82q]