use the model s=-1/2gt2+v0t+s0. An NBA basketball player has a 36-in.vertical leap. suppose that he jumped from ground level with an initial velocity of 8 under

Question

use the model s=-1/2gt²+v₀t+s_0.

An NBA basketball player has a 36-in.vertical leap. suppose that he jumped from ground level with an initial velocity of 8 under root 3 ft/sec.

 

a. write a model to express the player's height (in ft) above ground level t seconds after leaving the ground. Use exact values. do not round. 

 

 

Answer 1

First, we should convert that 36 inch height to feet, which is 3 feet.

 

Now, we should recognize that the variable s is a position variable in feet, V₀ is the initial velocity and s₀ is the initial position, which is 0, because the player starts on the ground. The goal of this problem is to find a value for 'g', and use that for the full equation for s(t).

 

Next, we take the derivative of the position function with respect to time

 

s(t) = -(1/2)gt² + V₀t +s₀

s'(t) = v)t) = -gt +V₀

 

What is his velocity at the maximum height? The vertical velocity of any object when it reaches the top is 0.

 

So now we plug that in, along with the initial velocity, and solve for t

 

0 = 8√3 - gt

 

gt = 8√3

t = 8√(3) / g

 

Now plug this time into the position function, recognizing that the max heigh the player jumped was 3 feet

 

s(t) = -(1/2) g (8√(3) / g)^2 + 8√3 * (8√(3) / g) = 3

 

    3 = -(1/2)*g*64*3/g² + 64*3/g

    3 = -(1/2)*64*3/g + 64*3/g             note that we have something like -(1/2)*A + A, which is equal to (1/2)A

    3 =  (1/2)*64*3/g                            divide both sides by 3

    1 =  (1/2)*64/g                                multiply both sides by g

    g =  (1/2)*64

    g = 32

 

So now we have our value for g, which is excellent, because g is just about 32 ft/s^2 in the real world. We can plug this back into the original s(t) equation, along with the initial velocity and initial position.

 

s(t) = -(1/2)*32*t² + 8√(3)*t + 0

s(t) = -16t² + 8√(3) * t 

 

Note that the right-most t is outside of the radical. It's tough to show that in this format. I hope this helps!