Answer:
vf = 30 m/s : (the magnitude of the velocity of the stone just before it hits the ground)
Explanation:
Because the stone moves with uniformly accelerated movement we apply the following formulas:
vf²=v₀²+2*g*h Formula (1)
Where:
h: displacement in meters (m)
v₀: initial speed in m/s
vf: final speed in m/s
g: acceleration due to gravity in m/s²
Free fall of the stone
Data
v₀ = 10 m/s
vf = 30.0 m/s
g = 9,8 m/s²
We replace data in the formula (1) to calculate h:
vf²=v₀²+2*g*h
(30)² = (10)² + (2)(9.8)*h
(30)²- (10)²= (2)(9.8)*h
h =( (30)²- (10)²) /( 2)(9.8)
h = 40.816 m
Semiparabolic movement of the stone
Data
v₀x = 10 m/s
v₀y = 0 m/s
g = 9.8 m/s²
h= 40.816 m
We replace data in the formula (1) to calculate vfy :
vfy² = v₀y² + 2*g*h
vfy² = 0 + (2)(9.8)( 40.816)
[tex]v_{fy}=\sqrt{2*9.8*40.816} = 28.284 \frac{m}{s}[/tex]
[tex]v_{f}=\sqrt{v_{ox}^2+v_{fy}^2}=\sqrt{(10)^2+(28.284)^2} = 30\frac{m}{s}[/tex]
The magnitude of the velocity of the stone just before it hits the ground is 30 m/s.
The given parameters;
initial vertical velocity of the stone, [tex]v_y_0[/tex] = 10 m/s
final vertical velocity of the stone, [tex]v_y_f[/tex] = 30 m/s
The height traveled by the stone before it hits the ground is calculated as;
[tex]v_y_f^2 = v_y_0^2 + 2gh\\\\h = \frac{v_y_f^2- v_y_0^2}{2g} \\\\h = \frac{(30)^2 - (10)^2}{2\times 9.8} \\\\h = 40.82 \ m[/tex]
If the the stone is projected horizontally with initial velocity of 10 m/s;
the initial vertical velocity = 0
Final vertical velocity of the stone is calculated as follow;
[tex]v_y_f^2 = v_y_0^2 + 2gh\\\\v_y_f^2 = 0 + 2\times 9.8\times 40.82\\\\v_y_f^2 = 800.07\\\\v_y_f = \sqrt{800.07} \\\\v_y_f = 28.28 \ m/s[/tex]
The horizontal velocity doesn't change.
the final horizontal velocity, [tex]v_x_f[/tex] = initial horizontal velocity = 10 m/s
The resultant of the final velocity of the stone before it hits the ground;
[tex]v _f= \sqrt{v_x_f^2 + v_y_f^2} \\\\v_f = \sqrt{10^2 + 28.28^2} \\\\v_f= 29.99 \ m/s \approx 30 \ m/s[/tex]
Thus, the magnitude of the velocity of the stone just before it hits the ground is 30 m/s.
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