Answer:
a) ![t=2.6\ s](https://tex.z-dn.net/?f=t%3D2.6%5C%20s)
b) ![s=43.7747\ m](https://tex.z-dn.net/?f=s%3D43.7747%5C%20m)
Explanation:
Given:
length of inclined roof, ![l=54\ m](https://tex.z-dn.net/?f=l%3D54%5C%20m)
inclination of roof below horizontal, ![\theta=17^{\circ}C](https://tex.z-dn.net/?f=%5Ctheta%3D17%5E%7B%5Ccirc%7DC)
acceleration of hammer on the roof, ![a_r=2.87\ m.s^{-2}](https://tex.z-dn.net/?f=a_r%3D2.87%5C%20m.s%5E%7B-2%7D)
height from the lower edge of the roof, ![h=46.5\ m](https://tex.z-dn.net/?f=h%3D46.5%5C%20m)
<u>Now, we find the final velocity when leaving the edge of the roof:</u>
Using the equation of motion:
![v^2=u^2+2.a_r.l](https://tex.z-dn.net/?f=v%5E2%3Du%5E2%2B2.a_r.l)
![v^2=0^2+2\times 2.87\times 54](https://tex.z-dn.net/?f=v%5E2%3D0%5E2%2B2%5Ctimes%202.87%5Ctimes%2054)
![v=17.6057\ m.s^{-1}](https://tex.z-dn.net/?f=v%3D17.6057%5C%20m.s%5E%7B-1%7D)
The direction of this velocity is 17° below the horizontal.
∴Vertical component of velocity:
![v_y=v.sin\ \theta](https://tex.z-dn.net/?f=v_y%3Dv.sin%5C%20%5Ctheta)
![v_y=17.6057\times sin\ 17^{\circ}](https://tex.z-dn.net/?f=v_y%3D17.6057%5Ctimes%20sin%5C%2017%5E%7B%5Ccirc%7D)
![v_y=5.1474\ m.s^{-1}](https://tex.z-dn.net/?f=v_y%3D5.1474%5C%20m.s%5E%7B-1%7D)
a.
<em><u>So, the time taken to fall on the ground:</u></em>
![h=ut+\frac{1}{2} g.t^2](https://tex.z-dn.net/?f=h%3Dut%2B%5Cfrac%7B1%7D%7B2%7D%20g.t%5E2)
here:
initial velocity, ![u=v_y=5.1474\ m.s^{-1}](https://tex.z-dn.net/?f=u%3Dv_y%3D5.1474%5C%20m.s%5E%7B-1%7D)
putting respective values
![46.5=5.1474\times t+0.5\times 9.8\times t^2](https://tex.z-dn.net/?f=46.5%3D5.1474%5Ctimes%20t%2B0.5%5Ctimes%209.8%5Ctimes%20t%5E2)
![t=2.6\ s](https://tex.z-dn.net/?f=t%3D2.6%5C%20s)
b.
Horizontal component of velocity, ![v_x=v.cos\ \theta=17.6057\ cos\ 17^{\circ}=16.8364\ m.s^{-1}](https://tex.z-dn.net/?f=v_x%3Dv.cos%5C%20%5Ctheta%3D17.6057%5C%20cos%5C%2017%5E%7B%5Ccirc%7D%3D16.8364%5C%20m.s%5E%7B-1%7D)
Since there is no air resistance so the horizontal velocity component remains constant.
<em><u>∴Horizontal distance from the edge of the roof where the hammer falls is given by:</u></em>
![s=v_x.t](https://tex.z-dn.net/?f=s%3Dv_x.t)
![s=16.8364\times 2.6](https://tex.z-dn.net/?f=s%3D16.8364%5Ctimes%202.6)
![s=43.7747\ m](https://tex.z-dn.net/?f=s%3D43.7747%5C%20m)