Answer:
The ball is in the air for 3.5 seconds
The initial horizontal component of velocity is 28.6 m/s
The vertical component of the final velocity is 34.3 m/s downward
The final velocity is 44.7 m/s in the direction 50.2° below the horizontal
Explanation:
A ball is thrown horizontally
That means the vertical component of the initial velocity [tex]u_{y}=0[/tex]
The initial velocity is the horizontal component [tex]u_{x}[/tex]
The ball is thrown from the top of a 60 m
That means the vertical displacement component y = 60 m
→ y = [tex]u_{y}[/tex] t + [tex]\frac{1}{2}[/tex] gt²
where g is the acceleration of gravity and t is the time
y = -60 m , g = -9.8 m/s² , [tex]u_{y}=0[/tex]
Substitute these values in the rule
→ -60 = 0 + [tex]\frac{1}{2}[/tex] (-9.8)t²
→ -60 = -4.9t²
Divide both sides by -4.9
→ 12.2449 = t²
Take √ for both sides
∴ t = 3.5 seconds
* The ball is in the air for 3.5 seconds
The initial velocity is the horizontal component [tex]u_{x}[/tex]
The ball lands 100 meter from the base of the building
That means the horizontal displacement x = 100 m
→ x = [tex]u_{x}[/tex] t
→ t = 3.5 s , x = 100 m
Substitute these values in the rule
→ 100 = [tex]u_{x}[/tex] (3.5)
Divide both sides by 3.5
→ [tex]u_{x}[/tex] = 28.57 m/s
The initial horizontal component of velocity is 28.6 m/s
The vertical component of the final velocity is [tex]v_{y}[/tex]
→ [tex]v_{y}[/tex] = [tex]u_{y}[/tex] + gt
→ [tex]u_{y}[/tex] = 0 , g = -9.8 m/s² , t = 3.5 s
Substitute these values in the rule
→ [tex]v_{y}[/tex] = 0 + (-9.8)(3.5)
→ [tex]v_{y}[/tex] = -34.3 m/s
The vertical component of the final velocity is 34.3 m/s downward
The final velocity v is the resultant vector of [tex]v_{x}[/tex] and [tex]v_{y}[/tex]
→ Its magnetude is [tex]v=\sqrt{(v_{x})^{2}+(v_{y})^{2}}[/tex]
→ Its direction [tex]tan^{-1}\frac{v_{y}}{v_{x}}[/tex]
→ [tex]v_{y}[/tex] = 28.6 , [tex]v_{y}[/tex] = -34.3
Substitute this values in the rules above
→ [tex]v=\sqrt{(28.6)^{2}+(-34.3)^{2}}=44.66[/tex]
→ Its direction [tex]tan^{-1}\frac{-34.3}{28.6}=-50.18[/tex]
The negative sign means the direction is below the horizontal
The final velocity is 44.7 m/s in the direction 50.2° below the horizontal
