A. The difference in the two ball's time in the air is 3 seconds
B. The velocity of each ball as it strikes the ground is 24.5 m/s
C. The balls 0.500 s after they are thrown are 14.7 m apart
<h3>Further explanation</h3>
Acceleration is rate of change of velocity.
![\large {\boxed {a = \frac{v - u}{t} } }](https://tex.z-dn.net/?f=%5Clarge%20%7B%5Cboxed%20%7Ba%20%3D%20%5Cfrac%7Bv%20-%20u%7D%7Bt%7D%20%7D%20%7D)
![\large {\boxed {d = \frac{v + u}{2}~t } }](https://tex.z-dn.net/?f=%5Clarge%20%7B%5Cboxed%20%7Bd%20%3D%20%5Cfrac%7Bv%20%2B%20u%7D%7B2%7D~t%20%7D%20%7D)
<em>a = acceleration ( m/s² )</em>
<em>v = final velocity ( m/s )</em>
<em>u = initial velocity ( m/s )</em>
<em>t = time taken ( s )</em>
<em>d = distance ( m )</em>
Let us now tackle the problem!
<u>Given:</u>
Initial Height = H = 19.6 m
Initial Velocity = u = 14.7 m/s
<u>Unknown:</u>
A. Δt = ?
B. v = ?
C. Δh = ?
<u>Solution:</u>
<h2>Question A:</h2><h3>First Ball</h3>
![h = H - ut - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=h%20%3D%20H%20-%20ut%20-%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2)
![0 = 19.6 - 14.7t - \frac{1}{2}(9.8)t^2](https://tex.z-dn.net/?f=0%20%3D%2019.6%20-%2014.7t%20-%20%5Cfrac%7B1%7D%7B2%7D%289.8%29t%5E2)
![0 = 19.6 - 14.7t - 4.9t^2](https://tex.z-dn.net/?f=0%20%3D%2019.6%20-%2014.7t%20-%204.9t%5E2)
![4.9t^2 + 14.7t - 19.6 = 0](https://tex.z-dn.net/?f=4.9t%5E2%20%2B%2014.7t%20-%2019.6%20%3D%200)
![t^2 + 3t - 4 = 0](https://tex.z-dn.net/?f=t%5E2%20%2B%203t%20-%204%20%3D%200)
![(t + 4)(t - 1) = 0](https://tex.z-dn.net/?f=%28t%20%2B%204%29%28t%20-%201%29%20%3D%200)
![(t - 1) = 0](https://tex.z-dn.net/?f=%28t%20-%201%29%20%3D%200)
![\boxed {t = 1 ~ second}](https://tex.z-dn.net/?f=%5Cboxed%20%7Bt%20%3D%201%20~%20second%7D)
<h3>Second Ball</h3>
![h = H + ut - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=h%20%3D%20H%20%2B%20ut%20-%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2)
![0 = 19.6 + 14.7t - \frac{1}{2}(9.8)t^2](https://tex.z-dn.net/?f=0%20%3D%2019.6%20%2B%2014.7t%20-%20%5Cfrac%7B1%7D%7B2%7D%289.8%29t%5E2)
![0 = 19.6 + 14.7t - 4.9t^2](https://tex.z-dn.net/?f=0%20%3D%2019.6%20%2B%2014.7t%20-%204.9t%5E2)
![4.9t^2 - 14.7t - 19.6 = 0](https://tex.z-dn.net/?f=4.9t%5E2%20-%2014.7t%20-%2019.6%20%3D%200)
![t^2 - 3t - 4 = 0](https://tex.z-dn.net/?f=t%5E2%20-%203t%20-%204%20%3D%200)
![(t - 4)(t + 1) = 0](https://tex.z-dn.net/?f=%28t%20-%204%29%28t%20%2B%201%29%20%3D%200)
![(t - 4) = 0](https://tex.z-dn.net/?f=%28t%20-%204%29%20%3D%200)
![\boxed {t = 4 ~ seconds}](https://tex.z-dn.net/?f=%5Cboxed%20%7Bt%20%3D%204%20~%20seconds%7D)
The difference in the two ball's time in the air is:
![\Delta t = 4 ~ seconds - 1 ~ second](https://tex.z-dn.net/?f=%5CDelta%20t%20%3D%204%20~%20seconds%20-%201%20~%20second)
![\large {\boxed {\Delta t = 3 ~ seconds} }](https://tex.z-dn.net/?f=%5Clarge%20%7B%5Cboxed%20%7B%5CDelta%20t%20%3D%203%20~%20seconds%7D%20%7D)
<h2>Question B:</h2><h3>First Ball</h3>
![v^2 = u^2 - 2gH](https://tex.z-dn.net/?f=v%5E2%20%3D%20u%5E2%20-%202gH)
![v^2 = (-14.7)^2 + 2(-9.8)(-19.6)](https://tex.z-dn.net/?f=v%5E2%20%3D%20%28-14.7%29%5E2%20%2B%202%28-9.8%29%28-19.6%29)
![v^2 = 600.25](https://tex.z-dn.net/?f=v%5E2%20%3D%20600.25)
![v = \sqrt {600.25}](https://tex.z-dn.net/?f=v%20%3D%20%5Csqrt%20%7B600.25%7D)
![\boxed {v = 24.5 ~ m/s}](https://tex.z-dn.net/?f=%5Cboxed%20%7Bv%20%3D%2024.5%20~%20m%2Fs%7D)
<h3>Second Ball</h3>
![v^2 = u^2 - 2gH](https://tex.z-dn.net/?f=v%5E2%20%3D%20u%5E2%20-%202gH)
![v^2 = (14.7)^2 + 2(-9.8)(-19.6)](https://tex.z-dn.net/?f=v%5E2%20%3D%20%2814.7%29%5E2%20%2B%202%28-9.8%29%28-19.6%29)
![v^2 = 600.25](https://tex.z-dn.net/?f=v%5E2%20%3D%20600.25)
![v = \sqrt {600.25}](https://tex.z-dn.net/?f=v%20%3D%20%5Csqrt%20%7B600.25%7D)
![\boxed {v = 24.5 ~ m/s}](https://tex.z-dn.net/?f=%5Cboxed%20%7Bv%20%3D%2024.5%20~%20m%2Fs%7D)
The velocity of each ball as it strikes the ground is 24.5 m/s
<h2>Question C:</h2><h3>First Ball</h3>
![h = H - ut - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=h%20%3D%20H%20-%20ut%20-%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2)
![h = 19.6 - 14.7(0.5) - \frac{1}{2}(9.8)(0.5)^2](https://tex.z-dn.net/?f=h%20%3D%2019.6%20-%2014.7%280.5%29%20-%20%5Cfrac%7B1%7D%7B2%7D%289.8%29%280.5%29%5E2)
![\boxed {h = 11.025 ~ m}](https://tex.z-dn.net/?f=%5Cboxed%20%7Bh%20%3D%2011.025%20~%20m%7D)
<h3>Second Ball</h3>
![h = H + ut - \frac{1}{2}gt^2](https://tex.z-dn.net/?f=h%20%3D%20H%20%2B%20ut%20-%20%5Cfrac%7B1%7D%7B2%7Dgt%5E2)
![h = 19.6 + 14.7(0.5) - \frac{1}{2}(9.8)(0.5)^2](https://tex.z-dn.net/?f=h%20%3D%2019.6%20%2B%2014.7%280.5%29%20-%20%5Cfrac%7B1%7D%7B2%7D%289.8%29%280.5%29%5E2)
![\boxed {h = 25.725 ~ m}](https://tex.z-dn.net/?f=%5Cboxed%20%7Bh%20%3D%2025.725%20~%20m%7D)
The difference in the two ball's height after 0.500 s is:
![\Delta h = 25.725 ~ m - 11.025 ~ m](https://tex.z-dn.net/?f=%5CDelta%20h%20%3D%2025.725%20~%20m%20-%2011.025%20~%20m)
![\large {\boxed {\Delta h = 14.7 ~ m} }](https://tex.z-dn.net/?f=%5Clarge%20%7B%5Cboxed%20%7B%5CDelta%20h%20%3D%2014.7%20~%20m%7D%20%7D)
<h3>Learn more</h3>
<h3>Answer details</h3>
Grade: High School
Subject: Physics
Chapter: Kinematics
Keywords: Velocity , Driver , Car , Deceleration , Acceleration , Obstacle