Answer:
a. ![\displaystyle k_o=1350\ J](https://tex.z-dn.net/?f=%5Cdisplaystyle%20k_o%3D1350%5C%20J)
b. ![\displaystyle k_1=0\ J](https://tex.z-dn.net/?f=%5Cdisplaystyle%20k_1%3D0%5C%20J)
c. ![\Delta k=-1350\ J](https://tex.z-dn.net/?f=%5CDelta%20k%3D-1350%5C%20J)
d. ![W=-1350\ J](https://tex.z-dn.net/?f=W%3D-1350%5C%20J)
e. ![F=-675\ N](https://tex.z-dn.net/?f=F%3D-675%5C%20N)
Explanation:
<u>Work and Kinetic Energy
</u>
When an object moves at a certain velocity v0 and changes it to v1, a change in its kinetic energy is achieved:
![\Delta k=k_1-k_0](https://tex.z-dn.net/?f=%5CDelta%20k%3Dk_1-k_0)
Knowing that
![\displaystyle k=\frac{mv^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20k%3D%5Cfrac%7Bmv%5E2%7D%7B2%7D)
We have
![\displaystyle \Delta k=\frac{mv_1^2}{2}-\frac{mv_0^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5CDelta%20k%3D%5Cfrac%7Bmv_1%5E2%7D%7B2%7D-%5Cfrac%7Bmv_0%5E2%7D%7B2%7D)
The work done by the force who caused the change of velocity (acceleration) is
![\displaystyle W=\frac{mv_1^2}{2}-\frac{mv_0^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20W%3D%5Cfrac%7Bmv_1%5E2%7D%7B2%7D-%5Cfrac%7Bmv_0%5E2%7D%7B2%7D)
If we know the distance x traveled by the object, the work can also be calculated by
![W=F.x](https://tex.z-dn.net/?f=W%3DF.x)
Being F the force responsible for the change of velocity
The 75 kg baseball player has an initial velocity of 6 m/s, then he slides and stops
a. Before the slide, his initial kinetic energy is
![\displaystyle k_o=\frac{mv_0^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20k_o%3D%5Cfrac%7Bmv_0%5E2%7D%7B2%7D)
![\displaystyle k_o=\frac{(75)6^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20k_o%3D%5Cfrac%7B%2875%296%5E2%7D%7B2%7D)
![\boxed{\displaystyle k_o=1350\ J}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20k_o%3D1350%5C%20J%7D)
b. Once he reaches the base, the player is at rest, thus his final kinetic energy is
![\displaystyle k_1=\frac{(75)0^2}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20k_1%3D%5Cfrac%7B%2875%290%5E2%7D%7B2%7D)
![\boxed{\displaystyle k_1=0\ J}](https://tex.z-dn.net/?f=%5Cboxed%7B%5Cdisplaystyle%20k_1%3D0%5C%20J%7D)
c. The change of kinetic energy is
![\Delta k=k_1-k_0=0\ J-1350\ J](https://tex.z-dn.net/?f=%5CDelta%20k%3Dk_1-k_0%3D0%5C%20J-1350%5C%20J)
![\boxed{\Delta k=-1350\ J}](https://tex.z-dn.net/?f=%5Cboxed%7B%5CDelta%20k%3D-1350%5C%20J%7D)
d. The work done by friction to stop the player is
![W=\Delta k=k_1-k_0](https://tex.z-dn.net/?f=W%3D%5CDelta%20k%3Dk_1-k_0)
![\boxed{W=-1350\ J}](https://tex.z-dn.net/?f=%5Cboxed%7BW%3D-1350%5C%20J%7D)
e. We compute the force of friction by using
![W=F.x](https://tex.z-dn.net/?f=W%3DF.x)
and solving for x
![\displaystyle F=\frac{W}{x}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%3D%5Cfrac%7BW%7D%7Bx%7D)
![\displaystyle F=\frac{-1350\ J}{2\ m}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20F%3D%5Cfrac%7B-1350%5C%20J%7D%7B2%5C%20m%7D)
![\boxed{F=-675\ N}](https://tex.z-dn.net/?f=%5Cboxed%7BF%3D-675%5C%20N%7D)
The negative sign indicates the force is against movement