A duck is thrown into the air with an initial velocity of 100ft/s ....

Question

I was given this formula: s=-16t²+v₀t+S₀

 

A rubber duckie is thrown into the air from the ground with an initial velocity of 100ft/s. Neglecting air resistaqnce, what is the formula for its height (in feet) above the ground seconds after being thrown? After how many seconds will duckie be 50ft above the ground? How long will it take to return to the ground?

Answer 1

v₀ = initial velocity = 100 ft/s

S₀ = initial height = 0  since it was thrown from the ground  

 

The formula for height above the ground t seconds after being thrown is 

 

s = -16t² + 100t  

 

To find when the duckie will be 50 ft above the ground, set s = 50 and solve for t

 

50 = -16t² + 100t

16t² -100t + 50 = 0

8t² - 50t + 25 = 0  

 

This doesn't factor so use the quadratic equation 

 

t = [50 +/- sqrt(2500 - 4(8)(25))] / 16

t = [50 +/- sqrt(1700)] / 16  

t = [50 +/- sqrt(4*25*17)] / 16

t = [50 +/- 10sqrt(17)] / 16

t = [25 +/- 5sqrt(17)] / 8 

 

t ~ 5.702 or t ~ 0.548 seconds 

 

To find when it returns to the ground set s = 0 and solve for t

 

0 = -16t² + 100t

4t² - 25t = 0

t(4t - 25) = 0

 

t = 0 or t = 25/4 = 6.25 seconds 

 

It was thrown at t = 0 so it will return to the ground at t = 6.25 seconds

 

Answer 2

Your initial velocity is 100 ft/s.  We can write this as a variable.

 

v₀ = 100ft/s

 

For the first question

 

Initial position is at ground level, before the duckie was thrown.  We can write this as a variable.

 

S₀ = 0 ft

 

So your height above the ground is

 

s = -16t² + 100t

 

 

For the second question

 

s = 50 ft

 

We will use the equation from previous part to solve for t.

 

s = -16t² + 100t

 

50 = -16t² + 100t

 

0 = -16t² + 100t - 50

 

Use the quadratic formula to solve to t.

 

 

Last Question

 

The ball returns to the ground from the height of 50 ft.  We need to solve for t at this instant.

 

s = 0 ft

S₀ = 50 ft

 

s = -16t² + 100t + S₀

 

0 = -16t² + 100t + 50

 

Use the quadratic formula to solve for t.