If a baseball player hits a baseball from 4 feet off the ground with an initial velocity of 64 feet per second, how long will it
take the baseball to hit the ground? Use the equation h = −16t2 + 64t + 4. 2 plus or minus square root of 17 end root over 2 quantity of 2 plus or minus square root of 17 all over 2 2 plus or minus 4 square root of 17 quantity of 16 plus or minus square root of 17 all over 2
This is the concept of trajectories; We are required to calculate the time taken for the base ball whose distant to reach the maximum height has been modeled by h=-16t^2+64t+4.2 took to hit the ground. Here we proceed as follows; At the time when the base ball hits the ground the height,h=0 Thus; -16t^2+64t+4.2=0 this is a quadratic equation, to solve the quadratic equation we use the formula; t=[-b+/- sqrt(b^2-4ac)]/(2a) where; a=-16,b=64, c=4.2 thus substituting the values in our formula we get: t=[-64+/-sqrt(64^2-4*(-16)*4.2)]/(-16*2) t=[-64+/- sqrt(4364.8)]/(-32) t=[-64+/-66.1]/(-32) t=4.1 or-0.1 thus the we take the positive value t=4.1 and we conclude that the time taken for the ball to hit the ground was 4.1 seconds
, so that
![\dfrac{\mathrm d^2y}{\mathrm dx^2}=\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1x\dfrac{\mathrm dy}{\mathrm dz}\right]=-\dfrac1{x^2}\dfrac{\mathrm dy}{\mathrm dz}+\dfrac1x\left(\dfrac1x\dfrac{\mathrm d^2y}{\mathrm dz^2}\right)=\dfrac1{x^2}\left(\dfrac{\mathrm d^2y}{\mathrm dz^2}-\dfrac{\mathrm dy}{\mathrm dz}\right)](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dx%5E2%7D%3D%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Cdfrac1x%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dz%7D%5Cright%5D%3D-%5Cdfrac1%7Bx%5E2%7D%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dz%7D%2B%5Cdfrac1x%5Cleft%28%5Cdfrac1x%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dz%5E2%7D%5Cright%29%3D%5Cdfrac1%7Bx%5E2%7D%5Cleft%28%5Cdfrac%7B%5Cmathrm%20d%5E2y%7D%7B%5Cmathrm%20dz%5E2%7D-%5Cdfrac%7B%5Cmathrm%20dy%7D%7B%5Cmathrm%20dz%7D%5Cright%29)
with roots at 
. This means the characteristic solution for 
is
, this is
